#392615

order	512
nilpotency class	3
isoclinism	isoclinic class of #32894
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2846
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14693
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3331 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342961, #72342967, #72342976
maximal normal subgroup of	#72342961, #72342967, #72342976
maximal quotient of	#72285816

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299811);