#6380611

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

2098 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6287807);