#6439343

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2282 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)(13,14,18,20)(15,25,16,22)(17,24,19,23)(21,27,28,26);(1,2)(3,4)(6,9)(10,12)(14,20)(21,28)(22,25)(23,24);(2,4)(13,15,18,16)(14,21,20,28)(17,26,19,27)(22,23,25,24);(2,4)(5,7,8,11)(6,10,9,12)(13,16,18,15)(14,22,20,25)(17,26,19,27)(21,23,28,24);(13,17,18,19)(14,23,20,24)(15,26,16,27)(21,25,28,22)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346539);