#238494

order	512
nilpotency class	3
isoclinism	isoclinic class of #232331
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1151 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=di,f^c=fh,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145690);