#238797

order	512
nilpotency class	3
isoclinism	isoclinic class of #234099
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

945 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)(7,11,16,20)(8,13,9,12)(15,18,17,19)(21,22)(23,26)(24,28)(25,27)(29,32)(30,36)(31,35)(33,34);(1,2)(3,4)(5,7)(6,11)(8,17)(9,15)(10,16)(12,18)(13,19)(14,20)(21,23,25,31)(22,26,27,35)(24,32,34,30)(28,29,33,36);(1,3,4,2)(6,12)(8,9)(11,18)(13,14)(15,17)(19,20)(21,22)(23,29)(24,33)(25,27)(26,32)(28,34)(30,35)(31,36);(5,8,10,9)(6,13,14,12)(7,15,16,17)(11,18,20,19)(22,27)(26,35)(28,33)(29,36)

Polycyclic presentation:

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

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, 145993);