#188798

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

892 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,29)(20,34,36,40)(21,26)(22,27)(23,28)(24,32)(25,35,41,37)(30,38,46,48)(31,39)(33,42)(43,45,47,44);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,21,32,23,33,22,31)(18,24,27,29,28,39,26,42)(20,30,35,45,36,46,37,44)(25,38,40,43,41,48,34,47);(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,34)(27,40)(28,41)(29,43)(31,44)(32,45)(33,46)(39,48)(42,47);(3,9)(6,12)(10,15)(13,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,36,37)(24,29,39,42)(25,34,41,40)(30,44,46,45)(38,43,48,47)

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=dhi,f^c=fh,g^c=gh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95994);