#304412

order	512
nilpotency class	3
isoclinism	isoclinic class of #272923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

814 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=eh,c^2=efh,d^2=ehi,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, 211608);