#6594324

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

2164 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=gi,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6501520);