#392885

order	512
nilpotency class	3
isoclinism	isoclinic class of #32884
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1422 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)(9,10,14,18,15,19,13,27)(11,20,22,16,23,30,28,31)(12,17,26,33,24,32,25,37)(21,34,36,29,35,40,38,39);(9,11,14,22,15,23,13,28)(10,16,18,30,19,31,27,20)(12,21,26,36,24,35,25,38)(17,29,33,40,32,39,37,34);(1,2,3,4)(9,12)(10,17)(11,21)(13,25)(14,26)(15,24)(16,29)(18,33)(19,32)(20,34)(22,36)(23,35)(27,37)(28,38)(30,40)(31,39);(6,8)(12,24)(17,32)(21,35)(25,26)(29,39)(33,37)(34,40)(36,38)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300081);