#269136

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	#36710	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1241 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)(7,13,9,11)(8,12)(10,14)(15,18,16,19)(17,20)(21,22,26,28,25,30,24,29)(23,27,33,34,31,35,32,36);(1,2)(3,4)(5,7,10,9)(6,11,14,13)(8,15,17,16)(12,18,20,19)(21,23,25,31)(22,27,30,35)(24,32,26,33)(28,34,29,36);(2,4)(5,8,10,17)(6,12,14,20)(7,15,9,16)(11,18,13,19)(22,28)(24,26)(27,34)(29,30)(32,33)(35,36);(5,9,10,7)(6,13,14,11)(8,16,17,15)(12,19,20,18)(23,31)(27,35)(32,33)(34,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=h,d^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, 176332);