#6379000

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂×#3449	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4798 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6286196);