#7487719

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309080
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

1170 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7394915);