#471049

order	512
nilpotency class	3
isoclinism	isoclinic class of #460131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2760
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

864 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)(7,15)(8,12)(9,21,22,13)(10,14)(11,18)(16,27)(17,24,32,28)(19,33,34,25)(20,26)(23,30)(29,35,31,36);(1,2,4,3)(5,7)(6,11,14,15)(8,16)(9,17)(10,18)(12,23,26,27)(13,24,21,28)(19,29)(20,30)(22,32)(25,35,33,36)(31,34);(1,2)(3,4)(5,8,10,20)(6,12,14,26)(7,16,18,30)(9,19,22,34)(11,23,15,27)(13,25,21,33)(17,31,32,29)(24,36,28,35);(1,3,4,2)(5,9)(6,13)(7,17)(8,19)(10,22)(11,24)(12,25)(14,21)(15,28)(16,29)(18,32)(20,34)(23,35)(26,33)(27,36)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285194, #72285479, #72285562
maximal normal subgroup of	#72285194, #72285479, #72285562

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378245);