#303271

order	512
nilpotency class	3
isoclinism	isoclinic class of #272923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

884 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,14,18,22)(12,15,20,23)(17,24,19,21)(25,26,30,35)(27,33,38,32)(28,39,37,40)(29,36,31,34);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,19)(13,18)(15,21)(16,22)(17,20)(23,24)(25,27)(26,32)(28,31)(29,37)(30,38)(33,35)(34,40)(36,39);(1,3,5,4)(2,7,8,6)(9,12)(10,15)(11,17,18,19)(13,20)(14,21,22,24)(16,23)(25,28,29,38,30,37,31,27)(26,33,34,40,35,32,36,39);(1,4,5,3)(2,7,8,6)(25,29,30,31)(26,34,35,36)(27,37,38,28)(32,40,33,39)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 210467);