#497857

order	512
nilpotency class	4
isoclinism	isoclinic class of #33856
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	D₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	D₄×#107
prime radical	C₂×#135	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

581 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=fh,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=de,e^b=ei,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225388, #72225401, #72225403, …
maximal normal subgroup of	#72225388, #72225401, #72225403, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405053);