#261387

order	512
nilpotency class	3
isoclinism	isoclinic class of #246079
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	#39375	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1156 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)(7,14)(8,18)(9,12)(10,13)(11,16)(15,20)(17,19)(21,22,25,28)(23,26,30,33)(24,27,31,35)(29,34,36,32);(1,2,4,3)(5,6)(7,14)(8,12)(9,18)(10,13)(11,17)(15,20)(16,19)(21,23)(22,26)(24,29)(25,30)(27,34)(28,33)(31,36)(32,35);(5,7,10,15)(6,11,13,19)(8,17,9,16)(12,20,18,14)(21,24,25,31)(22,27,28,35)(23,29,30,36)(26,32,33,34);(1,3,4,2)(7,15)(11,19)(14,20)(16,17)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=hi,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,f^c=fh,g^c=gh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 168583);