#8434547

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4211 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

(1,2)(3,10)(4,13)(5,14)(6,8)(7,9)(11,12)(15,16)(17,18,24,27)(19,28,23,25)(20,22,21,26)(29,31,30,32)(36,44)(40,47)(43,48)(45,46);(1,3,7,6)(2,8,9,10)(4,11,14,16)(5,12,13,15)(17,19,24,23)(18,25,27,28)(20,30,21,29)(22,31,26,32)(33,34,38,42)(35,37,41,39)(36,40,44,47)(43,45,48,46);(17,20,24,21)(18,26,27,22)(19,29,23,30)(25,31,28,32)(33,35)(34,39)(36,43)(37,42)(38,41)(40,46)(44,48)(45,47);(1,4)(2,5)(3,11)(6,16)(7,14)(8,12)(9,13)(10,15)(17,21,24,20)(18,22,27,26)(19,30,23,29)(25,32,28,31)(33,35)(34,37)(36,43)(38,41)(39,42)(40,45)(44,48)(46,47);(1,5)(2,4)(3,12)(6,15)(7,13)(8,11)(9,14)(10,16)(17,22)(18,20)(19,31)(21,27)(23,32)(24,26)(25,30)(28,29)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,48)(42,47);(1,6,7,3)(2,8,9,10)(4,11,14,16)(5,15,13,12)(17,23,24,19)(18,25,27,28)(20,30,21,29)(22,32,26,31)(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=ghi,c^2=h,d^2=h,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, 8341743);