#93071

order	512
nilpotency class	4
isoclinism	isoclinic class of #7092
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#331
commutator	C₄×C₈	C₄²
Frattini, cosocle	C₂×C₈²	C₂²
socle	C₂²	#1296
prime radical	C₂³	#332
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

161 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 161 conjugacy classes of subgroups — too many to draw. Only its 82 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 — 2 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=i,e^2=fgi,f^2=h,g^2=i,b^a=bc,c^a=cf,e^a=efgh,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225492, #72225511, #72225520
maximal normal subgroup of	#72225492, #72225511, #72225520
maximal quotient of	#72225489

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267);