#195258

order	512
nilpotency class	3
isoclinism	isoclinic class of #18284
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1214 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)(5,6,9,12,10,7,8,11)(13,14,18,22,19,24,17,23)(15,25,27,20,29,35,28,36)(16,30,32,21,33,37,31,38)(26,34,40,43,42,44,41,39);(1,2,3,4)(5,7,10,6)(8,11,9,12)(13,15)(14,20)(16,26)(17,27)(18,28)(19,29)(21,34)(22,25)(23,35)(24,36)(30,43)(31,40)(32,41)(33,42)(37,39)(38,44);(5,6,10,7)(8,12,9,11)(13,16,19,33)(14,21,24,38)(15,26,29,42)(17,31,18,32)(20,34,36,44)(22,37,23,30)(25,39,35,43)(27,40,28,41);(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)(34,39,44,43)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102454);