#125494

order	512
nilpotency class	4
isoclinism	isoclinic class of #125419
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1514
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

279 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 279 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2,4,6,5,3,7,8)(9,10,11,17,14,18,13,22)(12,15,20,16,21,23,19,24);(1,3)(2,5)(4,6)(7,8)(10,15)(11,13)(12,20)(16,17)(18,23)(19,21)(22,24);(2,3)(6,8)(9,11,14,13)(12,20,21,19)(15,23)(16,24)

Polycyclic presentation:

2^9|a^2=e,c^2=fgh,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,d^c=di,e^d=ei

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, 32690);