#194195

order	512
nilpotency class	3
isoclinism	isoclinic class of #18866
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1300 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=h,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=di,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fi,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, 101391);