#6584797

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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	#25139	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1911 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)(7,11)(8,12)(9,10)(13,14,18,22)(15,23,19,20)(16,25,27,24)(17,21,28,26);(1,2)(3,4)(6,10)(11,12)(13,15,18,19)(14,20,22,23)(16,26,27,21)(17,25,28,24);(2,4)(5,7,9,8)(6,11,10,12)(13,16,15,21,18,27,19,26)(14,17,20,24,22,28,23,25);(1,2)(3,4)(5,8,9,7)(6,12,10,11)(13,17,18,28)(14,21,22,26)(15,24,19,25)(16,20,27,23);(2,4)(16,27)(17,28)(21,26)(24,25)

Polycyclic presentation:

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

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, 6491993);