#6668091

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2758 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)(8,10)(9,12)(13,14)(15,20)(16,23)(17,19)(18,22)(21,27)(24,25)(26,28);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,19,22)(14,18,17,20)(16,23,26,28)(21,27,24,25);(2,4)(13,16,19,26)(14,21,17,24)(15,23,22,28)(18,27,20,25);(6,10)(9,11)(13,17,19,14)(15,18,22,20)(16,24,26,21)(23,27,28,25);(2,4)(7,12)(9,11)(13,18,19,20)(14,22,17,15)(16,25,26,27)(21,23,24,28)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6575287);