#6377603

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#36652	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cf,e^b=ei,d^c=di,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6284799);