#6381017

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

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

2028 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6288213);