#238652

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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

887 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)(7,8)(9,15)(10,13)(11,14)(12,19)(16,18)(17,20)(21,22,25,29)(23,26,30,35)(24,27,33,28)(31,34,32,36);(1,2)(3,4)(5,7)(6,8)(9,17)(10,12)(11,16)(13,19)(14,18)(15,20)(21,23)(22,26)(24,32)(25,30)(27,36)(28,34)(29,35)(31,33);(1,3,4,2)(5,8)(6,12)(7,15)(9,20)(10,19)(11,18)(13,16)(14,17)(22,27)(24,33)(26,34)(28,29)(31,32)(35,36);(7,16)(8,18)(12,17)(19,20)(23,30)(26,35)(31,32)(34,36)

Polycyclic presentation:

2^9|a^2=i,c^2=e,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,d^c=dhi,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, 145848);