#395248

order	512
nilpotency class	3
isoclinism	isoclinic class of #395248
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14722
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1675 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)(4,10)(5,23)(6,27)(7,12)(8,13)(9,17)(11,25)(15,34)(16,44)(18,32)(19,47)(20,35)(21,37)(22,38)(24,39)(26,53)(28,33)(29,36)(30,42)(31,54)(40,46)(41,58)(43,57)(45,51)(48,64)(49,60)(50,52)(55,59)(56,61)(62,63);(1,3,6,17,8,18,7,28)(2,9,12,14,13,33,27,32)(4,19,21,15,22,42,29,43)(5,16,24,45,25,46,26,54)(10,34,36,47,38,57,37,30)(11,31,39,40,23,51,53,44)(20,48,49,41,50,61,55,62)(35,58,59,64,52,63,60,56);(1,4)(2,10)(3,15)(5,20)(6,21)(7,29)(8,22)(9,30)(11,35)(12,36)(13,38)(14,34)(16,41)(17,42)(18,43)(19,28)(23,52)(24,49)(25,50)(26,55)(27,37)(31,56)(32,57)(33,47)(39,59)(40,58)(44,63)(45,61)(46,62)(48,54)(51,64)(53,60);(1,5,6,24,8,25,7,26)(2,11,12,39,13,23,27,53)(3,16,17,45,18,46,28,54)(4,20,21,49,22,50,29,55)(9,31,14,40,33,51,32,44)(10,35,36,59,38,52,37,60)(15,41,42,61,43,62,19,48)(30,56,34,58,47,64,57,63)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318385, #72318935, #72318981
maximal normal subgroup of	#72318385, #72318935, #72318981

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302444);