#7135161

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1447 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=hi,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,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, 7042357);