#6437970

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437969
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	C₂²×C₄	C₂⁵
prime radical	#62852	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4712 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310916, #72311055, #72311064, …
maximal normal subgroup of	#72310916, #72311055, #72311064, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345166);