#125440

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	C₂³	#418
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 257 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=fg,c^2=fg,d^2=g,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=efg,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, 32636);