#376776

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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

1294 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=f,d^2=h,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, 283972);