#368240

order	512
nilpotency class	3
isoclinism	isoclinic class of #23996
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1584 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=eg,d^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,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, 275436);