#195109

order	512
nilpotency class	3
isoclinism	isoclinic class of #18376
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1026 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=i,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dhi,f^a=fhi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fh,g^c=gi

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, 102305);