#367220

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1233 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 274416);