#7233336

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
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

1362 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)(4,5)(7,8)(9,10)(11,17)(12,20)(13,15)(14,18)(16,22)(19,23)(21,24)(25,26)(27,33)(28,35)(29,44)(30,37)(31,36)(32,38)(34,43)(39,54)(40,53)(41,50)(42,48)(45,52)(46,49)(47,56)(51,55);(2,5)(3,7)(9,11,13,18)(10,14,15,17)(12,21,22,19)(16,24,20,23)(25,27,32,41)(26,33,38,50)(28,42,29,39)(30,46,31,40)(34,51,52,47)(35,48,44,54)(36,53,37,49)(43,55,45,56);(11,18)(14,17)(19,21)(23,24)(25,28,32,29)(26,34,38,52)(27,39,41,42)(30,43,31,45)(33,47,50,51)(35,37,44,36)(40,55,46,56)(48,53,54,49);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,20)(23,24)(25,29,32,28)(26,35,38,44)(27,39,41,42)(30,43,31,45)(33,48,50,54)(34,37,52,36)(40,56,46,55)(47,53,51,49);(2,5)(6,8)(9,12,13,22)(10,16,15,20)(11,19,18,21)(14,23,17,24)(25,30,32,31)(26,36,38,37)(27,40,41,46)(28,43,29,45)(33,49,50,53)(34,35,52,44)(39,55,42,56)(47,48,51,54)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=efi,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7140532);