#8505625

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

4043 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8412821);