#6668631

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2780 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=f,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,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6575827);