#6466005

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2126 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373201);