#493824

order	512
nilpotency class	4
isoclinism	isoclinic class of #33618
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#466
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1808
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

933 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,14)(4,21,24,35)(5,25)(6,12)(7,29)(8,13)(9,18)(10,23,34,28)(11,15)(16,43,44,62)(17,38)(19,40)(20,50)(22,33,53,60)(26,36)(27,55)(30,57,58,64)(31,47)(32,48)(37,46,61,49)(39,41)(42,52,56,54)(45,51,63,59);(1,3,15,41,7,20,27,47,8,17,26,48,6,19,5,18)(2,9,25,40,12,32,36,38,13,31,55,50,29,39,11,14)(4,16,42,57,23,46,53,63,24,44,56,64,28,49,22,45)(10,30,54,62,35,59,60,61,34,58,52,43,21,51,33,37);(1,4)(2,10)(3,16)(5,22)(6,28)(7,23)(8,24)(9,30)(11,33)(12,35)(13,34)(14,37)(15,42)(17,44)(18,45)(19,49)(20,46)(21,29)(25,54)(26,56)(27,53)(31,58)(32,59)(36,60)(38,61)(39,51)(40,62)(41,57)(43,50)(47,63)(48,64)(52,55);(3,17)(9,31)(14,38)(16,44)(18,47)(19,20)(30,58)(32,39)(37,61)(40,50)(41,48)(43,62)(45,63)(46,49)(51,59)(57,64)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,f^c=fi,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314256, #72314278
maximal normal subgroup of	#72314256, #72314278

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401020);