#480766

order	512
nilpotency class	4
isoclinism	isoclinic class of #480662
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

838 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 838 conjugacy classes of subgroups — too many to draw. Only its 198 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=hi,d^2=i,f^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=ch,d^b=di,e^b=ei,f^b=fg,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387962);