#6445161

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2115 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:

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

Polycyclic presentation:

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

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, 6352357);