#6378183

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2615	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2010 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)(6,9)(7,11)(13,14,18,22,17,19,15,20)(16,21,25,28,24,26,23,27);(5,6)(7,10)(8,9)(11,12)(14,19)(20,22)(21,26)(27,28);(2,4)(6,9)(10,12);(5,7,8,11)(6,10,9,12)(13,15,17,18)(14,20,19,22)(16,23,24,25)(21,27,26,28);(13,16,18,25,17,24,15,23)(14,21,22,28,19,26,20,27)

Polycyclic presentation:

2^9|a^2=h,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=cfi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6285379);