#395433

order	512
nilpotency class	3
isoclinism	isoclinic class of #395432
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

3365 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^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,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	#72318509, #72318546, #72318571
maximal normal subgroup of	#72318509, #72318546, #72318571

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302629);