#9851092

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	#54601	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3971 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=ghi,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=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, 9758288);