#7500182

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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1364 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)(27,38)(28,44)(29,35)(30,36)(31,37)(32,47)(33,42)(34,45)(39,49)(40,53)(41,54)(43,50)(46,51)(48,56)(52,55);(2,5)(3,7)(9,10,13,14)(11,17,18,15)(12,21,20,16)(19,22,24,23)(25,27,31,42)(26,33,37,38)(28,39,29,40)(30,41,32,43)(34,48,51,52)(35,49,44,53)(36,50,47,54)(45,55,46,56);(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,28,31,29)(26,34,37,51)(27,39,42,40)(30,45,32,46)(33,48,38,52)(35,47,44,36)(41,55,43,56)(49,54,53,50);(1,3,4,7)(2,6,5,8)(11,18)(15,17)(19,24)(22,23)(25,29,31,28)(26,35,37,44)(27,40,42,39)(30,45,32,46)(33,49,38,53)(34,47,51,36)(41,55,43,56)(48,54,52,50);(2,5)(6,8)(9,12,13,20)(10,16,14,21)(11,19,18,24)(15,22,17,23)(25,30,31,32)(26,36,37,47)(27,41,42,43)(28,45,29,46)(33,50,38,54)(34,35,51,44)(39,55,40,56)(48,49,52,53)

Polycyclic presentation:

2^9|a^2=fhi,b^2=hi,c^2=i,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=efi,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, 7407378);