#7223789

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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1226 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)(27,39)(28,35)(29,45)(30,38)(31,37)(32,36)(33,42)(34,44)(40,49)(41,50)(43,51)(46,53)(47,52)(48,56)(54,55);(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)(26,33)(28,40)(29,46)(30,41)(31,42)(32,43)(34,48)(35,53)(36,50)(37,39)(38,51)(44,55)(45,49)(47,56)(52,54);(2,5)(6,8)(25,28,31,29)(26,34,37,52)(27,40,42,46)(30,44,32,47)(33,48,39,54)(35,38,45,36)(41,55,43,56)(49,50,53,51);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,29,31,28)(26,35,37,45)(27,40,42,46)(30,44,32,47)(33,49,39,53)(34,38,52,36)(41,56,43,55)(48,50,54,51);(2,5)(6,8)(9,13,14,12)(10,16,17,22)(11,19,20,21)(15,24,18,23)(25,30,31,32)(26,36,37,38)(27,41,42,43)(28,44,29,47)(33,50,39,51)(34,35,52,45)(40,55,46,56)(48,53,54,49)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,d^2=fh,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dhi,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, 7130985);