#405307

order	512
nilpotency class	3
isoclinism	isoclinic class of #394273
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1438 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312503);