#7300242

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234345
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=i,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,g^c=gi,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, 7207438);