#6437996

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437995
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4879 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260844, #72267077, #72267082, …
maximal normal subgroup of	#72260844, #72267077, #72267082, …
maximal quotient of	#72261079

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345192);