#395435

order	512
nilpotency class	3
isoclinism	isoclinic class of #395434
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14724
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3586	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3367 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)(4,21)(5,27,29,11)(6,13)(7,12)(8,14)(9,18)(10,25)(16,43)(17,42,48,59)(19,34)(20,33)(22,41)(23,52,54,50)(24,38)(26,37)(28,39,30,40)(31,46)(32,47,56,49)(35,45)(36,53,57,55)(44,62,63,58)(51,61,64,60);(1,3)(2,9)(4,22)(5,17)(6,19)(7,18)(8,20)(10,35)(11,32)(12,33)(13,15)(14,34)(16,25)(21,43)(23,51)(24,46)(26,45)(27,56)(28,49)(29,48)(30,47)(31,37)(36,60)(38,41)(39,42)(40,59)(44,53)(50,62)(52,58)(54,64)(55,63)(57,61);(1,4,8,26)(2,10,14,38)(3,16,20,46)(5,23,29,54)(6,25,7,24)(9,31,34,43)(11,36,27,57)(12,21,13,37)(15,41,33,35)(17,44,48,63)(18,22,19,45)(28,55,30,53)(32,58,56,62)(39,52,40,50)(42,61,59,60)(47,64,49,51);(1,5)(2,11)(3,17)(4,23)(6,30)(7,28)(8,29)(9,32)(10,36)(12,40)(13,39)(14,27)(15,42)(16,44)(18,49)(19,47)(20,48)(21,50)(22,51)(24,55)(25,53)(26,54)(31,58)(33,59)(34,56)(35,60)(37,52)(38,57)(41,61)(43,62)(45,64)(46,63)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306456, #72306459, #72306465
maximal normal subgroup of	#72306456, #72306459, #72306465

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302631);