#6473324

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2550 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6380520);