#403994

order	512
nilpotency class	3
isoclinism	isoclinic class of #393982
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23400	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2562 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311190);