#7232006

order	512
nilpotency class	3
isoclinism	isoclinic class of #7207866
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×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1184 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=fghi,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7139202);