#238485

order	512
nilpotency class	3
isoclinism	isoclinic class of #232253
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	#37556	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1135 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)(7,8)(9,15)(10,14)(11,13)(12,19)(16,20)(17,18)(21,22)(23,26)(24,27)(25,28)(29,33)(30,32)(31,34)(35,36);(1,2)(3,4)(5,7)(6,8)(9,16)(10,12)(11,17)(13,18)(14,19)(15,20)(21,23)(22,26)(24,29)(25,30)(27,33)(28,32)(31,35)(34,36);(1,3,4,2)(5,8)(6,12)(7,15)(9,20)(10,19)(11,18)(13,16)(14,17)(26,32)(33,36);(6,13)(8,18)(14,15)(19,20)(21,24,25,31)(22,27,28,34)(23,29,30,35)(26,33,32,36)

Polycyclic presentation:

2^9|c^2=e,d^2=i,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=dh,f^c=fh,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145681);