#6467151

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437412
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×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2230 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)(13,14,17,19)(15,21,22,28)(16,24,26,20)(18,25,27,23);(2,3)(13,15)(14,18)(16,25)(17,22)(19,27)(20,28)(21,24)(23,26);(1,2,4,3)(6,9)(10,12)(14,19)(15,22)(20,24)(23,25);(1,3,4,2)(15,22)(18,27)(21,28)(23,25);(5,7,8,11)(6,10,9,12)(14,19)(18,27)(20,24)(21,28)

Polycyclic presentation:

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

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, 6374347);