#7233343

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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1348 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)(25,26,31,37)(27,38,42,33)(28,35,29,44)(30,36,32,47)(34,45,53,43)(39,54,40,49)(41,51,46,50)(48,56,52,55);(2,5)(3,7)(9,11,13,18)(10,14,15,17)(12,21,22,19)(16,24,20,23)(25,27,31,42)(26,33,37,38)(28,40,29,39)(30,46,32,41)(34,52,53,48)(35,54,44,49)(36,51,47,50)(43,55,45,56);(11,18)(14,17)(19,21)(23,24)(25,28,31,29)(26,34,37,53)(27,39,42,40)(30,43,32,45)(33,48,38,52)(35,36,44,47)(41,55,46,56)(49,51,54,50);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,20)(23,24)(25,29,31,28)(26,35,37,44)(27,40,42,39)(30,43,32,45)(33,49,38,54)(34,36,53,47)(41,55,46,56)(48,51,52,50);(2,5)(6,8)(9,12,13,22)(10,16,15,20)(11,19,18,21)(14,23,17,24)(25,30,31,32)(26,36,37,47)(27,41,42,46)(28,43,29,45)(33,50,38,51)(34,44,53,35)(39,55,40,56)(48,49,52,54)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=fhi,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, 7140539);