#6545008

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2120 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)(13,14,18,22,19,23,17,21)(15,24,26,16,25,27,28,20);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(15,25)(16,20)(24,27)(26,28);(6,9)(10,12)(13,15)(14,20)(16,23)(17,26)(18,28)(19,25)(21,24)(22,27);(5,7)(6,10)(8,11)(9,12)(13,16,19,20)(14,15,23,25)(17,24,18,27)(21,28,22,26);(7,11)(10,12)(13,17,19,18)(14,21,23,22)(15,26,25,28)(16,27,20,24)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6452204);