#393019

order	512
nilpotency class	3
isoclinism	isoclinic class of #32944
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1818 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=fi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,e^b=ei,h^b=hi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300215);