#394548

order	512
nilpotency class	3
isoclinism	isoclinic class of #394064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#35088	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2383 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)(7,11)(8,12)(9,13)(10,14)(15,24)(16,22)(17,23)(18,31)(19,26)(20,27)(21,29)(25,32)(28,33)(30,34)(35,36);(1,2)(3,4)(5,7)(6,11)(8,15)(9,16)(10,17)(12,24)(13,22)(14,23)(18,28)(19,29)(20,30)(21,26)(25,35)(27,34)(31,33)(32,36);(1,3,4,2)(5,8)(6,12)(7,15)(9,18)(10,19)(11,21)(13,25)(14,26)(16,28)(17,29)(20,32)(22,33)(23,24)(27,31)(30,36)(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|c^2=e,d^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,f^b=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285165, #72285355, #72285495
maximal normal subgroup of	#72285165, #72285355, #72285495

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301744);