#7006584

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2099 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:

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

Polycyclic presentation:

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

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, 6913780);