#304012

order	512
nilpotency class	3
isoclinism	isoclinic class of #268355
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	C₂×#547	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

916 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)(9,10,14,17)(11,18,21,15)(12,23,16,13)(19,22,24,20)(25,26)(27,33)(28,31)(29,32)(30,35)(34,38)(36,40)(37,39);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,24)(13,20)(14,21)(16,19)(17,18)(22,23)(25,27,28,35)(26,30,31,33)(29,34,37,36)(32,40,39,38);(1,3,5,4)(2,7,8,6)(9,12,13,10,14,16,23,17)(11,19,22,15,21,24,20,18)(26,31)(27,34)(29,37)(30,38)(33,40)(35,36);(1,4,5,3)(2,7,8,6)(9,13,14,23)(10,16,17,12)(11,20,21,22)(15,19,18,24)(26,31)(30,33)(32,39)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 211208);