#7515524

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
prime radical	#25109	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1837 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^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,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,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, 7422720);