#400054

order	512
nilpotency class	3
isoclinism	isoclinic class of #395191
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2431
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14795
prime radical	#23414	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1364 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)(3,15)(4,10)(5,24)(6,14)(7,12)(8,13)(9,18)(11,25)(16,42)(17,30)(19,32)(20,31)(21,50)(22,39)(23,36)(26,38)(27,37)(28,35)(29,49)(33,44)(34,53)(40,45)(41,46)(43,55)(47,57)(48,56)(51,59)(52,54)(58,64)(60,62)(61,63);(1,3)(2,9)(4,16)(5,17)(6,20)(7,18)(8,19)(10,33)(11,30)(12,32)(13,31)(14,15)(21,43)(22,49)(23,45)(24,41)(25,48)(26,47)(27,46)(28,44)(29,36)(34,58)(35,40)(37,57)(38,56)(39,42)(50,60)(51,63)(52,64)(53,62)(54,61)(55,59);(1,4)(2,10)(3,16)(5,21)(6,22)(7,28)(8,23)(9,29)(11,34)(12,35)(13,36)(14,39)(15,40)(17,43)(18,44)(19,45)(20,49)(24,54)(25,51)(26,52)(27,53)(30,55)(31,33)(32,42)(37,59)(38,50)(41,60)(46,62)(47,64)(48,63)(56,61)(57,58);(1,5,6,27,8,26,7,25)(2,11,12,38,13,37,14,24)(3,17,20,46,19,47,18,48)(4,21,22,53,23,52,28,51)(9,30,32,56,31,57,15,41)(10,34,35,50,36,59,39,54)(16,43,49,62,45,64,44,63)(29,55,42,61,33,58,40,60)

Polycyclic presentation:

2^9|d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285525, #72285536, #72286087, …
maximal normal subgroup of	#72285525, #72285536, #72286087, …
maximal quotient of	C₂×#400054

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 307250);