#7517959

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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1849 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)(15,25,29,20)(16,30,32,38)(17,24,19,22)(21,34,39,33)(26,42,44,37)(27,40,28,36)(31,41,43,35);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,18,29)(14,20,23,25)(16,31,32,43)(17,28,19,27)(21,37,39,42)(22,36,24,40)(26,34,44,33)(30,35,38,41);(13,16)(14,21)(15,26)(17,33)(18,32)(19,34)(20,35)(22,30)(23,39)(24,38)(25,41)(27,31)(28,43)(29,44)(36,42)(37,40);(2,4)(5,8,9,7)(6,10,11,12)(14,22)(15,27)(16,32)(17,19)(21,38)(23,24)(26,43)(28,29)(30,39)(31,44)(35,41)(36,40);(6,11)(10,12)(13,17,18,19)(14,22,23,24)(15,28,29,27)(16,33,32,34)(20,36,25,40)(21,30,39,38)(26,43,44,31)(35,42,41,37)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=fi,e^2=i,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,d^c=di,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, 7425155);