#7199735

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1194 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)(4,8)(5,7)(9,10)(11,15)(12,16)(13,21)(14,17)(18,20)(19,23)(22,24)(25,26,31,32,30,36,29,35)(27,33,43,45,38,48,39,49)(28,34,44,46,42,52,40,51)(37,47,54,55,41,50,53,56);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,18)(21,24)(26,32)(28,40)(29,31)(33,45)(35,36)(37,53)(39,43)(41,54)(42,44)(46,51)(48,49)(55,56);(1,4,5,3)(2,6,7,8)(9,10)(11,18)(12,21)(13,16)(14,17)(15,20)(19,23)(22,24)(25,27)(26,33)(28,41)(29,39)(30,38)(31,43)(32,45)(34,50)(35,49)(36,48)(37,42)(40,54)(44,53)(46,56)(47,52)(51,55);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,28,31,44,30,42,29,40)(26,34,32,46,36,52,35,51)(27,37,43,54,38,41,39,53)(33,47,45,55,48,50,49,56);(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(27,38)(33,48)(37,41)(39,43)(45,49)(47,50)(53,54)(55,56)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=g,d^2=fhi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7106931);