#8430448

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	#52343	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4140 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,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, 8337644);