#7145781

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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

2253 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)(3,4)(7,11)(10,12)(13,14,17,22,18,21,25,24)(15,19,23,26,20,28,27,16);(6,9)(7,11)(13,15,18,20)(14,16,21,26)(17,27,25,23)(19,22,28,24);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,24)(17,23)(18,20)(21,28)(22,26)(25,27);(6,9)(10,12)(13,16,25,28,18,26,17,19)(14,20,24,23,21,15,22,27);(5,7,8,11)(6,10,9,12)(14,21)(15,20)(22,24)(23,27)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=fi,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7052977);