#125427

order	512
nilpotency class	4
isoclinism	isoclinic class of #125425
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1555
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

339 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 339 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,c^2=fgi,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fi,g^a=gi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,e^c=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, 32623);