#269039

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×#534	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1103 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=h,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, 176235);