#239564

order	512
nilpotency class	3
isoclinism	isoclinic class of #232331
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	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1212 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)(7,10)(8,11)(9,13)(12,16)(14,18)(15,17)(19,20)(21,22)(23,26)(24,27)(25,28)(29,34)(30,33)(31,35)(32,36);(1,2)(3,4)(5,7,9,14)(6,10,13,18)(8,15,16,20)(11,17,12,19)(21,23)(22,26)(24,29)(25,30)(27,34)(28,33)(31,36)(32,35);(1,3,4,2)(6,11)(8,16)(10,17)(12,13)(15,20)(18,19)(21,24)(22,27)(23,29)(25,31)(26,32)(28,35)(30,36)(33,34);(7,14)(10,18)(15,20)(17,19)(23,30)(24,31)(26,33)(27,35)

Polycyclic presentation:

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

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, 146760);