#6445698

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437748
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
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3266 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=gi,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=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, 6352894);