#354912

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁵	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1574 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=eh,e^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,c^b=ci,d^b=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262108);