#6443299

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439029
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4891 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341489, #72341493, #72341496
maximal normal subgroup of	#72341489, #72341493, #72341496

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350495);