#269054

order	512
nilpotency class	3
isoclinism	isoclinic class of #269019
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	C₂³×C₄	C₂⁴
prime radical	C₂×#1600	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,f^b=fh,e^c=eh,f^c=fi,f^d=fh

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, 176250);