#6432334

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	#23390	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2390 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339530);