#331013

order	512
nilpotency class	3
isoclinism	isoclinic class of #331007
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3150	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

533 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=h,d^2=fi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,g^a=gh,c^b=cg,e^b=eh,g^b=gi,d^c=de,e^c=eh,f^c=fh,g^c=gi,g^d=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238209);