#261479

order	512
nilpotency class	3
isoclinism	isoclinic class of #246239
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#39420	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1183 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)(11,15)(16,19)(17,20)(21,22)(23,29)(24,27)(25,28)(26,31)(30,33)(32,35)(34,36);(1,2,4,3)(5,6)(7,14)(8,12)(9,18)(10,13)(11,16)(15,19)(17,20)(21,23,25,31)(22,26,28,29)(24,30,32,36)(27,34,35,33);(5,7,10,17)(6,11,13,19)(8,16,9,15)(12,20,18,14)(21,24)(22,27)(23,30)(25,32)(26,33)(28,35)(29,34)(31,36);(1,3,4,2)(22,28)(24,32)(26,29)(30,36)

Polycyclic presentation:

2^9|b^2=e,c^2=h,d^2=ei,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=di,f^c=fh,g^c=gh

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, 168675);