#6454142

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2067 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:

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

Polycyclic presentation:

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

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, 6361338);