#269130

order	512
nilpotency class	3
isoclinism	isoclinic class of #269019
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	C₂³×C₄	C₂⁴
prime radical	C₂×#538	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1022 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)(7,15,18,11)(8,12)(9,16,20,13)(10,14)(17,19)(21,22,27,35,26,31,25,30)(23,28,24,34,33,29,32,36);(1,2)(3,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,20)(13,19)(14,15)(21,23,26,33)(22,28,31,29)(24,25,32,27)(30,36,35,34);(2,4)(5,8,10,19)(6,12,14,17)(7,16,18,13)(9,11,20,15)(21,24,26,32)(22,29,31,28)(23,25,33,27)(30,36,35,34);(5,9,10,20)(6,13,14,16)(7,17,18,12)(8,11,19,15)(21,25,26,27)(22,30,31,35)(23,24,33,32)(28,34,29,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=hi,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, 176326);