#6565758

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533087
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	#2867	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1879 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(5,6,10,9)(7,11,13,15)(8,12,16,17)(14,18,19,20)(21,22,26,29)(23,30,32,35)(24,28,25,33)(27,34,36,31);(1,2,3,4)(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,19)(17,20);(7,13)(11,15)(14,19)(18,20)(21,23,26,32)(22,27,29,36)(24,31,25,34)(28,35,33,30);(5,8,10,16)(6,12,9,17)(7,14,13,19)(11,18,15,20)(21,24,26,25)(22,28,29,33)(23,31,32,34)(27,35,36,30);(5,9,10,6)(7,15,13,11)(8,12,16,17)(14,18,19,20)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=i,d^2=hi,e^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,g^c=gi,e^d=eh

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6472954);