#6431917

order	512
nilpotency class	3
isoclinism	isoclinic class of #23444
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
socle	C₂²	C₂²×#120
prime radical	C₂×#3449	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4878 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280733, #72281147, #72282332, …
maximal normal subgroup of	#72280733, #72281147, #72282332, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339113);