#6685612

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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

2036 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)(7,10)(8,9)(11,12)(13,14,17,18)(15,19,21,25)(16,20,24,23)(22,26,27,28);(6,9)(7,11)(14,18)(15,19)(16,23)(20,24)(21,25)(22,27);(1,2,3,4)(13,15)(14,19)(16,22)(17,21)(18,25)(20,26)(23,28)(24,27);(6,9)(10,12)(15,21)(19,25)(22,27)(26,28);(5,7,8,11)(6,10,9,12)(13,16,17,24)(14,20,18,23)(15,22,21,27)(19,26,25,28)

Polycyclic presentation:

2^9|a^2=i,c^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

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, 6592808);