#323178

order	512
nilpotency class	3
isoclinism	isoclinic class of #323133
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3205	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

712 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=eg,c^2=hi,d^2=eh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,f^c=fi,g^c=ghi

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, 230374);