#331025

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	#3205	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

539 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:

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

Polycyclic presentation:

2^9|a^2=ei,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, 238221);