#6465881

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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	#28488	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2013 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)(7,10)(8,9)(11,12)(13,14,19,16)(15,22,23,28)(17,21,18,27)(20,25,24,26);(2,3)(6,9)(10,12)(15,23)(20,24)(22,28)(25,26);(1,2,4,3)(13,15,18,26,19,23,17,25)(14,20,21,22,16,24,27,28);(1,3,4,2)(13,16,19,14)(15,24,23,20)(17,21,18,27)(22,26,28,25);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,21,16,27)(15,25,23,26)(20,22,24,28)

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373077);