#7223758

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1210 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)(9,10,14,17)(11,18,20,15)(12,16,13,22)(19,24,21,23)(25,26,32,37)(27,33,41,51)(28,35,29,43)(30,36,31,46)(34,44,53,42)(38,48,39,54)(40,49,45,50)(47,55,52,56);(1,2)(3,8)(4,5)(6,7)(9,11,14,20)(10,15,17,18)(12,21,13,19)(16,24,22,23)(25,27,32,41)(26,33,37,51)(28,39,29,38)(30,45,31,40)(34,52,53,47)(35,54,43,48)(36,50,46,49)(42,55,44,56);(2,5)(6,8)(25,28,32,29)(26,34,37,53)(27,38,41,39)(30,42,31,44)(33,47,51,52)(35,36,43,46)(40,55,45,56)(48,50,54,49);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,29,32,28)(26,35,37,43)(27,39,41,38)(30,42,31,44)(33,48,51,54)(34,36,53,46)(40,55,45,56)(47,50,52,49);(2,5)(6,8)(9,13,14,12)(10,16,17,22)(11,19,20,21)(15,24,18,23)(25,30,32,31)(26,36,37,46)(27,40,41,45)(28,42,29,44)(33,49,51,50)(34,43,53,35)(38,55,39,56)(47,48,52,54)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,d^2=fh,e^2=fi,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,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7130954);