#6373018

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
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	C₂×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2382 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)(13,14,19,23,18,22,16,21)(15,24,27,28,26,20,25,17);(6,9)(10,12)(13,15)(14,20)(16,25)(17,23)(18,26)(19,27)(21,28)(22,24);(2,4)(5,7,8,11)(6,10,9,12)(13,16,18,19)(14,21,22,23)(15,25,26,27)(17,20,28,24);(13,17,19,24,18,28,16,20)(14,15,23,27,22,26,21,25);(14,22)(15,26)(21,23)(25,27)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280214);