#7272069

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234321
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1200 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18,21,24)(19,22,26,30)(20,23,28,32)(25,29,27,31)(33,34)(35,42)(36,40)(37,46)(38,41)(39,44)(43,47)(45,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,19,21,26)(18,22,24,30)(20,27,28,25)(23,31,32,29);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(17,20)(18,23)(19,25)(21,28)(22,29)(24,32)(26,27)(30,31)(33,35,37,44,38,45,36,43)(34,39,40,42,41,47,46,48);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,16,10,15)(20,28)(23,32)(25,27)(29,31)(33,34)(35,39)(36,46)(37,40)(38,41)(42,44)(43,48)(45,47);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(19,26)(22,30)(25,27)(29,31)(33,36,38,37)(34,40,41,46)(35,43,45,44)(39,42,47,48)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,d^2=f,e^2=fi,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=dh,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7179265);