#6370033

order	512
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂⁴
commutator	C₂²	C₂³×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂⁴×C₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277229);