#298374

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#1656	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

998 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)(5,6,9,12)(7,14,15,11)(8,18,13,10)(16,20,17,19)(21,22)(23,29)(24,26)(25,33)(27,31)(28,32)(30,35)(34,36);(1,2,4,3)(5,7)(6,11)(8,19)(9,15)(10,16)(12,14)(13,20)(17,18)(22,26)(23,30)(25,32)(27,34);(5,8,10,6,9,13,18,12)(7,14,16,19,15,11,17,20)(21,23,25,31,24,30,32,36)(22,27,28,29,26,34,33,35);(1,3,4,2)(6,12)(8,13)(11,14)(19,20)(22,26)(27,34)(28,33)(29,35)

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,c^2=f,d^2=eh,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dhi,g^c=gh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205570);