#7488810

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300916
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	#45267	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1684 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:

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

Polycyclic presentation:

2^9|c^2=f,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7396006);