#6369727

order	512
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄³ · Schur multiplier (H₂) = C₂⁶×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂⁴
commutator	C₂	C₂²×C₄³
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4362 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(9,10,13,17,14,18,24,30)(11,15,21,27,20,26,33,37)(12,16,23,29,22,28,34,38)(19,25,32,36,31,35,39,40);(1,2,3,4)(9,11)(10,15)(12,19)(13,21)(14,20)(16,25)(17,27)(18,26)(22,31)(23,32)(24,33)(28,35)(29,36)(30,37)(34,39)(38,40);(5,6,7,8)(9,12)(10,16)(11,19)(13,23)(14,22)(15,25)(17,29)(18,28)(20,31)(21,32)(24,34)(26,35)(27,36)(30,38)(33,39)(37,40);(12,22)(16,28)(19,31)(23,34)(25,35)(29,38)(32,39)(36,40);(11,20)(15,26)(19,31)(21,33)(25,35)(27,37)(32,39)(36,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,f^2=i,e^b=ei,d^c=di

Conjugacy classes & character table

272 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276923);