#7486901

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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1163 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,24)(21,25)(23,26)(27,30)(28,31)(29,32)(33,34,37,41,38,40,46,48)(35,42,43,44,39,47,45,36);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,21,26)(18,22,25,23)(20,28,29,27)(24,31,32,30)(35,39)(36,44)(42,47)(43,45);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,29)(30,32)(33,35)(34,36)(37,45)(38,39)(40,44)(41,47)(42,48)(43,46);(3,11)(6,9)(10,16)(13,15)(19,26)(22,23)(27,28)(30,31)(33,36)(34,39)(35,40)(37,42)(38,44)(41,45)(43,48)(46,47);(2,8)(6,9)(7,14)(13,15)(17,20,21,29)(18,24,25,32)(19,27,26,28)(22,30,23,31)(34,40)(36,44)(41,48)(42,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ch,d^b=dfh,e^b=ehi,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, 7394097);