#7522448

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2209 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)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,19,24)(15,20,29,37)(16,21,31,39)(17,22,18,23)(25,33,42,44)(26,34,41,43)(27,35,28,36)(30,38,32,40);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,20)(16,25)(17,28)(18,27)(19,29)(21,33)(22,36)(23,35)(24,37)(26,30)(31,42)(32,41)(34,38)(39,44)(40,43);(2,4)(13,16)(14,21)(15,25)(17,32)(18,30)(19,31)(20,33)(22,40)(23,38)(24,39)(26,27)(28,41)(29,42)(34,35)(36,43)(37,44);(2,4)(5,7,9,8)(6,11,12,10)(13,16)(14,21)(15,26)(17,30)(18,32)(19,31)(20,34)(22,38)(23,40)(24,39)(25,27)(28,42)(29,41)(33,35)(36,44)(37,43);(5,8,9,7)(6,11,12,10)(13,17,19,18)(14,22,24,23)(15,27,29,28)(16,30,31,32)(20,35,37,36)(21,38,39,40)(25,41,42,26)(33,43,44,34)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7429644);