#7500302

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300916
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

1156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 64 points:

(1,2,6,10)(3,11,13,7)(4,8,5,9)(12,16,14,15)(18,22)(20,28)(23,25)(27,32)(33,34,38,43)(35,39,44,53)(36,41,48,58)(37,40,50,42)(45,52,47,56)(46,54,59,51)(49,55,62,57)(60,64,61,63);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,16)(10,11)(17,18,21,22)(19,25,26,23)(20,29,28,24)(27,30,32,31)(33,35)(34,39)(36,47)(37,46)(38,44)(40,54)(41,56)(42,51)(43,53)(45,48)(49,61)(50,59)(52,58)(55,63)(57,64)(60,62);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,19,21,26)(18,23,22,25)(20,27,28,32)(24,30,29,31)(34,40)(35,44)(36,48)(37,50)(39,51)(41,57)(42,43)(52,64)(53,54)(55,58)(56,63)(60,61);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16)(19,26)(23,25)(27,32)(30,31)(33,36)(34,41)(35,45)(37,49)(38,48)(39,52)(40,55)(42,57)(43,58)(44,47)(46,60)(50,62)(51,63)(53,56)(54,64)(59,61);(2,10)(7,11)(8,9)(15,16)(17,20,21,28)(18,24,22,29)(19,27,26,32)(23,30,25,31)(35,44)(36,48)(39,53)(41,58)(46,59)(49,62)(51,54)(55,57)

Polycyclic presentation:

2^9|a^2=fh,b^2=hi,c^2=f,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,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, 7407498);