#7333329

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324564
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1277 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=gi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7240525);