#500213

order	512
nilpotency class	4
isoclinism	isoclinic class of #499855
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	C₂²×#41	D₄
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1003 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 1003 conjugacy classes of subgroups — too many to draw. Only its 134 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)(3,6)(4,5)(7,8)(9,10,13,15)(11,19,21,16)(12,23,25,35)(14,28,27,18)(17,30,31,26)(20,36,37,40)(22,33,29,34)(24,39,32,38);(2,5)(6,8)(9,11)(10,16)(12,24)(13,21)(14,29)(15,19)(17,20)(18,34)(22,27)(23,38)(25,32)(26,36)(28,33)(30,40)(31,37)(35,39);(1,3,4,7)(2,6,5,8)(9,12,13,25)(10,17,15,31)(11,20,21,37)(14,30,27,26)(16,32,19,24)(18,35,28,23)(22,39,29,38)(33,36,34,40);(2,5)(6,8)(11,21)(16,19)(20,37)(22,29)(24,32)(33,34)(36,40)(38,39)

Polycyclic presentation:

2^9|a^2=i,c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fi,g^a=ghi,c^b=cg,d^b=di,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, 407409);