#7574667

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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

1312 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=i,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7481863);