#6459801

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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

3165 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)(13,14,20,24)(15,17,22,28)(16,18,23,19)(21,26,27,25);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,19,26,20,22,18,25)(14,21,23,28,24,27,16,17);(6,10)(7,11)(13,16,19,14,20,23,18,24)(15,17,26,21,22,28,25,27);(7,11)(9,12)(13,17)(14,22)(15,24)(16,26)(18,27)(19,21)(20,28)(23,25);(13,18,20,19)(14,23,24,16)(15,25,22,26)(17,21,28,27)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gh,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,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, 6366997);