#512764

order	512
nilpotency class	5
isoclinism	isoclinic class of #33982
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator	C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×#2263	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

443 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2,4,3)(5,6)(7,13)(8,12)(9,26)(10,29)(11,17)(14,22)(15,23)(16,27)(18,34)(19,30)(20,48)(21,49)(24,41)(25,33)(28,55)(31,51)(32,50)(35,45)(36,46)(37,44)(38,54)(39,52)(40,53)(42,56)(43,66)(47,62)(57,67)(58,65)(59,63)(60,64)(61,68);(6,12)(8,22)(9,23)(10,27)(13,30)(14,15)(16,41)(17,33)(19,45)(20,46)(21,44)(24,54)(25,52)(26,31)(28,53)(29,32)(34,56)(35,36)(37,62)(38,55)(39,40)(42,63)(43,64)(47,68)(48,57)(49,58)(50,51)(59,60)(61,66)(65,67);(1,3,4,2)(5,7)(6,13)(8,19)(9,20)(10,21)(11,18)(12,30)(14,35)(15,36)(16,37)(17,34)(22,45)(23,46)(24,47)(25,42)(26,48)(27,44)(28,43)(29,49)(31,57)(32,58)(33,56)(38,61)(39,59)(40,60)(41,62)(50,65)(51,67)(52,63)(53,64)(54,68)(55,66);(7,18)(13,34)(19,42)(20,43)(21,44)(30,56)(35,59)(36,60)(37,49)(45,63)(46,64)(47,65)(48,66)(57,61)(58,62)(67,68)

Polycyclic presentation:

2^9|a^2=f,c^2=f,e^2=gh,g^2=hi,h^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,e^b=eg,g^b=gh,h^b=hi,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232007, #72322373, #72322380, …
maximal normal subgroup of	#72232007, #72322373, #72322380, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419960);