#494380

order	512
nilpotency class	4
isoclinism	isoclinic class of #33551
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×D₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	C₂×#1867
prime radical	#3449	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1066 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,h^a=hi,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	#72281842, #72283959, #72312293, …
maximal normal subgroup of	#72281842, #72283959, #72312293, …
maximal quotient of	C₂×#494380

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401576);