#7589212

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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1542 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,22,26)(19,23,28,37)(20,24,32,41)(21,25,35,44)(27,36,45,47)(29,38,33,42)(30,39,46,48)(31,40,34,43);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,28)(18,23,26,37)(20,30,32,46)(21,33,35,29)(24,39,41,48)(25,42,44,38)(27,34,45,31)(36,43,47,40);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19)(18,23)(20,27)(21,29)(22,28)(24,36)(25,38)(26,37)(30,31)(32,45)(33,35)(34,46)(39,40)(41,47)(42,44)(43,48);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,20,22,32)(18,24,26,41)(19,27,28,45)(21,34,35,31)(23,36,37,47)(25,43,44,40)(29,46,33,30)(38,48,42,39);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(19,28)(23,37)(27,45)(29,33)(30,46)(36,47)(38,42)(39,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=fgi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,e^b=ei,h^b=hi,e^c=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, 7496408);