#7522655

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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
prime radical	#36009	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2200 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7429851);