#303806

order	512
nilpotency class	3
isoclinism	isoclinic class of #271131
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	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

819 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=efh,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=dhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 211002);