#8443745

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3941 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8350941);