#481831

order	512
nilpotency class	4
isoclinism	isoclinic class of #11799
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
socle	C₂²	C₂²×#99
prime radical	#34633	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1075 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,d^2=i,e^2=h,f^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,c^b=ci,e^b=eh,f^b=fgh,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389027);