#6630327

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629016
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
prime radical	#2847	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2193 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)(7,10)(8,9)(11,12)(13,14,18,20)(15,24,25,28)(16,22,21,27)(17,23,19,26);(6,9)(10,12)(14,20)(15,25)(22,27)(23,26);(13,15,19,22,18,25,17,27)(14,21,23,24,20,16,26,28);(2,4)(13,16)(14,22)(15,26)(17,24)(18,21)(19,28)(20,27)(23,25);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,23,20,26)(15,27,25,22)(16,28,21,24)

Polycyclic presentation:

2^9|a^2=i,c^2=g,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ci,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6537523);