#8437829

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	#2631	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3954 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=ghi,d^2=gh,e^2=h,f^2=h,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, 8345025);