#122982

order	512
nilpotency class	4
isoclinism	isoclinic class of #122971
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator, Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
prime radical	C₂⁵	D₈
LCS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

644 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 46 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=fh,f^2=h,b^a=bd,c^a=ce,d^a=df,f^a=fh,g^a=gi,d^b=df,e^b=egi,f^b=fh,d^c=dg,e^d=ei

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30178);