#492999

order	512
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#99
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	C₂×C₄×#99
prime radical	#3449	C₂²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1191 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,f^2=i,b^a=be,d^a=di,g^a=gh,e^b=eh,h^b=hi,g^e=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281046, #72283959, #72312289, …
maximal normal subgroup of	#72281046, #72283959, #72312289, …
maximal quotient of	C₂×#492999

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400195);