#6552318

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	C₂×#550	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1904 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ei,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6459514);