#6532612

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499761
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×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 ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4368 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)(5,6)(7,9)(8,11)(10,12)(15,21)(18,22)(23,24)(26,27);(6,9)(7,12)(10,11)(13,14)(15,22)(16,19)(17,20)(18,21)(23,27)(24,26)(25,28);(2,4)(13,15,17,21)(14,18,20,22)(16,24,25,23)(19,27,28,26);(13,16,17,25)(14,19,20,28)(15,23,21,24)(18,26,22,27);(14,20)(18,22)(19,28)(26,27)

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dh

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, 6439808);