#238634

order	512
nilpotency class	3
isoclinism	isoclinic class of #233668
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	C₂×C₄²	C₂⁴
prime radical	#37606	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1033 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)(7,11)(8,12)(9,13)(10,14)(15,19)(16,18)(17,20)(21,22)(23,30)(24,28)(25,33)(26,29)(27,31)(32,36)(34,35);(1,2)(3,4)(5,7)(6,11)(8,17)(9,15)(10,16)(12,20)(13,19)(14,18)(24,25)(27,35)(28,33)(31,34);(1,3,4,2)(6,12)(7,15)(8,9)(13,14)(16,17)(19,20)(21,23)(22,27)(24,31)(25,34)(26,32)(28,36)(29,35)(30,33);(7,16)(11,18)(15,17)(19,20)(21,24,26,25)(22,28,29,33)(23,31,32,34)(27,36,35,30)

Polycyclic presentation:

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

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, 145830);