#123404

order	512
nilpotency class	4
isoclinism	isoclinic class of #123403
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²	#1712
commutator, Frattini, cosocle	C₈²	C₂³
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

230 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 48 points:

(2,6)(3,7)(4,10)(8,16)(9,14)(11,15)(12,13)(17,18,22,26)(19,27,31,43)(20,32,34,47)(21,35,33,25)(23,36,38,29)(24,39,40,28)(30,46,45,42)(37,44,41,48);(17,19,28,44,21,30,34,36,22,31,39,48,33,45,20,29)(18,23,32,42,25,37,40,43,26,38,47,46,35,41,24,27);(1,2,7,13,4,8,11,14,5,9,15,16,10,12,3,6)

Polycyclic presentation:

2^9|a^2=h,b^2=df,c^2=eg,d^2=fh,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,g^a=gi

Conjugacy classes & character table

134 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216236, #72216268, #72216272, …
maximal normal subgroup of	#72216236, #72216268, #72216272, …
maximal quotient of	#72216272

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30600);