#238602

order	512
nilpotency class	3
isoclinism	isoclinic class of #233642
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	#37554	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1059 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145798);