#7520822

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1794 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=fi,d^2=i,e^2=fi,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, 7428018);