#6432627

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431710
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2914 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,d^2=fg,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,f^b=fi,g^b=gi,f^c=fi,g^c=gi,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, 6339823);