#8492901

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3882 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=h,e^2=gi,f^2=g,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, 8400097);