#7500155

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1348 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(1,2,4,5)(3,6,7,8)(10,14)(12,20)(15,17)(19,24)(25,26,31,37)(27,39,43,33)(28,35,29,46)(30,38,32,36)(34,47,52,45)(40,54,41,49)(42,50,44,51)(48,55,53,56);(2,5)(3,7)(9,10,13,14)(11,17,18,15)(12,21,20,16)(19,22,24,23)(25,27,31,43)(26,33,37,39)(28,40,29,41)(30,42,32,44)(34,48,52,53)(35,49,46,54)(36,50,38,51)(45,55,47,56);(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,28,31,29)(26,34,37,52)(27,40,43,41)(30,45,32,47)(33,48,39,53)(35,38,46,36)(42,55,44,56)(49,51,54,50);(1,3,4,7)(2,6,5,8)(11,18)(15,17)(19,24)(22,23)(25,29,31,28)(26,35,37,46)(27,41,43,40)(30,45,32,47)(33,49,39,54)(34,38,52,36)(42,55,44,56)(48,51,53,50);(2,5)(6,8)(9,12,13,20)(10,16,14,21)(11,19,18,24)(15,22,17,23)(25,30,31,32)(26,36,37,38)(27,42,43,44)(28,45,29,47)(33,50,39,51)(34,35,52,46)(40,55,41,56)(48,49,53,54)

Polycyclic presentation:

2^9|a^2=fh,b^2=hi,c^2=i,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7407351);