#6682433

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682420
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 ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2492 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284627, #72284666, #72284808
maximal normal subgroup of	#72284627, #72284666, #72284808

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6589629);