#196217

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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×#386
prime radical	#60702	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1058 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=dh,e^b=eh,f^b=fh,g^b=gh,d^c=di,e^c=eh,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, 103413);