#365167

order	512
nilpotency class	3
isoclinism	isoclinic class of #24464
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1881 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)(7,16)(8,17,22,30)(9,23)(10,14)(11,15)(12,21,28,20)(13,24)(18,26)(19,27,33,36)(25,29)(31,34,35,32);(1,2)(3,4)(5,7,10,6,11,14,16,15)(8,17,20,12,22,30,21,28)(9,18,25,13,26,23,24,29)(19,31,34,27,33,35,32,36);(2,4)(5,8)(6,12)(7,17)(9,19)(10,20)(11,22)(13,27)(14,30)(15,28)(16,21)(18,31)(23,35)(24,32)(25,34)(26,33)(29,36);(5,9)(6,13)(7,18)(8,19)(10,25)(11,26)(12,27)(14,23)(15,29)(16,24)(17,31)(20,34)(21,32)(22,33)(28,36)(30,35)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272363);