#7300355

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234485
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

1135 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)(33,34,39,43)(35,40,38,47)(36,45,37,42)(41,46,48,44);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(19,25)(20,27)(22,28)(23,30)(33,35,39,38)(34,40,43,47)(36,46,37,44)(41,45,48,42);(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,36,39,37)(34,41,43,48)(35,44,38,46)(40,42,47,45);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(19,25)(22,28)(26,31)(29,32)(33,37,39,36)(34,42,43,45)(35,44,38,46)(40,41,47,48);(2,8)(6,9)(7,16)(14,15)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)(36,37)(41,48)(42,45)(44,46)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=i,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=ehi,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, 7207551);