#500371

order	512
nilpotency class	4
isoclinism	isoclinic class of #499867
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

849 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 849 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,f^c=fi,g^c=gi,h^c=hi,g^f=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407567);