#483960

order	512
nilpotency class	4
isoclinism	isoclinic class of #483956
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂³	#442
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

407 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 407 conjugacy classes of subgroups — too many to draw. Only its 166 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,9,6,3,10,8)(4,11,12,15,13,7,14,16)(17,18,22,27,23,28,20,25)(19,29,32,46,33,47,30,44)(21,35,38,45,37,43,34,26)(24,39,40,31,42,36,41,48);(1,3)(2,6)(4,11)(5,8)(7,13)(9,10)(12,15)(14,16)(17,19,23,33)(18,24,28,42)(20,32,22,30)(21,36,37,39)(25,40,27,41)(26,44,45,46)(29,35,47,43)(31,34,48,38);(1,4,10,14,6,13,5,12)(2,7,8,15,3,11,9,16)(18,25,28,27)(21,37)(24,40,42,41)(26,35,45,43)(29,44,47,46)(31,48)(34,38)(36,39);(2,3)(7,11)(8,9)(15,16)(17,20,23,22)(18,25,28,27)(19,30,33,32)(21,34,37,38)(24,41,42,40)(26,43,45,35)(29,44,47,46)(31,39,48,36)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=i,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=dhi,e^a=eg,g^a=gi,d^b=dh,e^b=egi,f^b=fi,g^b=gi,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225272, #72225276, #72225279, …
maximal normal subgroup of	#72225272, #72225276, #72225279, …
maximal quotient of	#72343707, #72343758

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391156);