#195332

order	512
nilpotency class	3
isoclinism	isoclinic class of #18346
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1228 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=hi,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=di,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102528);