#494397

order	512
nilpotency class	4
isoclinism	isoclinic class of #33568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×B(4,𝔽₂)
commutator	C₂×Q₈	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	C₂×#1867
prime radical	#3449	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1052 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,e^b=ei,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317352, #72317395, #72317420
maximal normal subgroup of	#72317352, #72317395, #72317420

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401593);