#6432330

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431707
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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2336 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)(7,10)(8,9)(11,12)(13,14,19,24,18,23,17,22)(15,25,26,21,16,20,27,28);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,18)(17,27)(19,26)(21,22)(23,25)(24,28);(2,4)(13,16,18,15)(14,21,23,28)(17,27,19,26)(20,24,25,22);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,22,23,24)(15,26,16,27)(20,28,25,21);(14,23)(20,25)(21,28)(22,24)

Polycyclic presentation:

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

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, 6339526);