#8458098

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	#54866	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4096 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)(17,18,23,27)(19,24,30,28)(20,22,26,32)(21,31,29,25)(33,34,40,38)(35,44,46,41)(36,47,42,37)(39,43,45,48);(1,3,7,6)(2,8,9,10)(4,11,14,16)(5,12,13,15)(17,18,23,27)(19,28,30,24)(20,22,26,32)(21,25,29,31)(33,35)(34,41)(36,45)(37,48)(38,44)(39,42)(40,46)(43,47);(17,19,23,30)(18,24,27,28)(20,29,26,21)(22,25,32,31)(33,36)(34,37)(35,45)(38,47)(39,46)(40,42)(41,48)(43,44);(1,4)(2,5)(3,11)(6,16)(7,14)(8,12)(9,13)(10,15)(17,20)(18,22)(19,29)(21,30)(23,26)(24,25)(27,32)(28,31)(33,37,40,47)(34,42,38,36)(35,43,46,48)(39,41,45,44);(1,5)(2,4)(3,12)(6,15)(7,13)(8,11)(9,14)(10,16)(17,21,23,29)(18,25,27,31)(19,20,30,26)(22,24,32,28)(33,38,40,34)(35,44,46,41)(36,47,42,37)(39,48,45,43);(1,6,7,3)(2,8,9,10)(4,11,14,16)(5,15,13,12)(17,22)(18,26)(19,25)(20,27)(21,24)(23,32)(28,29)(30,31)(33,39)(34,43)(35,42)(36,46)(37,44)(38,48)(40,45)(41,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=i,d^2=h,e^2=hi,f^2=gh,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, 8365294);