#461967

order	512
nilpotency class	3
isoclinism	isoclinic class of #453981
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

884 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=egi,b^2=fi,c^2=ef,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 369163);