#7224968

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1199 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=gi,d^2=h,e^2=f,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, 7132164);