#374353

order	512
nilpotency class	3
isoclinism	isoclinic class of #196002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1742 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fhi,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,f^c=fi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281549);