#7504063

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500661
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#31026	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1963 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(5,6)(7,8)(9,11)(10,12)(13,14,18,24,19,23,15,20)(16,27,25,22,17,28,26,21);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,19,18)(14,20,23,24)(16,25,17,26)(21,27,22,28);(2,4)(13,16,19,17)(14,21,23,22)(15,25,18,26)(20,27,24,28);(6,11)(7,12)(13,17,19,16)(14,22,23,21)(15,26,18,25)(20,28,24,27);(5,8,9,10)(6,7,11,12)(14,23)(20,24)(21,22)(27,28)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,g^c=gi,g^d=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308154, #72308353
maximal normal subgroup of	#72308154, #72308353

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7411259);