#6432372

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431710
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	#23421	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2324 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339568);