#7234262

order	512
nilpotency class	3
isoclinism	isoclinic class of #7207830
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1191 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)(3,6)(4,7)(5,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26)(27,37)(28,34)(29,36)(30,32)(31,35)(33,41)(38,53)(39,46)(40,49)(42,47)(43,56)(44,52)(45,51)(48,54)(50,55);(3,4)(6,7)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(26,32)(28,42)(33,46)(35,36)(37,49)(38,43)(44,45)(54,55);(1,3,5,4)(2,6,8,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,27,29,39,30,40,31,41)(26,33,35,49,32,46,36,37)(28,43,45,54,42,38,44,55)(34,50,52,53,47,48,51,56);(1,3,5,4)(2,7,8,6)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28)(26,34)(27,38)(29,45)(30,42)(31,44)(32,47)(33,48)(35,52)(36,51)(37,53)(39,55)(40,43)(41,54)(46,50)(49,56);(1,4,5,3)(2,7,8,6)(10,18)(15,19)(16,17)(23,24)(25,29,30,31)(26,35,32,36)(27,39,40,41)(28,44,42,45)(33,49,46,37)(34,51,47,52)(38,54,43,55)(48,53,50,56)

Polycyclic presentation:

2^9|b^2=fi,c^2=fgh,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=di,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7141458);