#364556

order	512
nilpotency class	3
isoclinism	isoclinic class of #23912
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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×#3018	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2360 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:

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

Polycyclic presentation:

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

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, 271752);