#480115

order	512
nilpotency class	4
isoclinism	isoclinic class of #2127
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
socle	C₂²	C₂²×#99
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1051 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=i,e^2=i,f^2=h,b^a=be,e^a=eg,g^a=gi,d^b=di,e^b=ei,f^b=fgi,f^e=fi

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, 387311);