#360203

order	512
nilpotency class	3
isoclinism	isoclinic class of #24805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	C₂×#380
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1620 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)(7,18)(8,22,25,31)(9,13)(10,26)(11,16)(12,17)(14,24,32,27)(15,21)(19,28,34,36)(20,29)(23,30,35,33);(1,2)(3,4)(5,7,10,9,12,21,11,20)(6,13,16,18,17,29,26,15)(8,19,24,23,25,34,27,35)(14,28,31,33,32,36,22,30);(2,4)(5,8)(6,14)(7,19)(9,23)(10,24)(11,27)(12,25)(13,28)(15,30)(16,31)(17,32)(18,33)(20,35)(21,34)(22,26)(29,36);(5,9,10,21,12,20,11,7)(6,15,16,13,17,18,26,29)(8,23,24,34,25,35,27,19)(14,30,31,28,32,33,22,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267399);