#6370422

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator	C₂²	C₂³×C₄²
socle	C₂³	C₂⁴×C₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=g,g^2=i,b^a=bf,c^a=ci,d^a=df,e^a=ei,c^b=cfi,d^c=di,e^d=ei

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277618);