#469877

order	512
nilpotency class	3
isoclinism	isoclinic class of #460096
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

948 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)(3,4)(5,6)(7,11)(8,12)(9,13,21,29)(10,14)(15,22)(16,23,33,36)(17,24)(18,28,20,26)(19,27)(25,31)(30,35,32,34);(5,7,10,17)(6,11,14,24)(8,15,19,31)(9,16,21,33)(12,25,27,22)(13,23,29,36)(18,30,20,32)(26,35,28,34);(5,8)(6,12)(7,15)(9,20,21,18)(10,19)(11,22)(13,28,29,26)(14,27)(16,32,33,30)(17,31)(23,35,36,34)(24,25);(2,4)(5,9)(6,13,14,29)(7,16)(8,18)(10,21)(11,23,24,36)(12,26,27,28)(15,30)(17,33)(19,20)(22,34,25,35)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284731, #72284765, #72284889
maximal normal subgroup of	#72284731, #72284765, #72284889

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 377073);