#6570596

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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 ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2218 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)(5,6,11,10)(7,8,9,12)(14,18)(16,19)(20,24)(21,25)(23,26)(27,28);(5,7,11,9)(6,12,10,8)(13,14,19,24,17,20,16,18)(15,21,26,28,22,27,23,25);(1,2,3,4)(5,8,11,12)(6,7,10,9)(14,20)(15,22)(18,24)(23,26);(5,9,11,7)(6,12,10,8)(13,15)(14,21)(16,23)(17,22)(18,25)(19,26)(20,27)(24,28);(5,10,11,6)(7,8,9,12)(14,20)(15,22)(18,24)(23,26)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6477792);