#8445953

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	#40392	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4034 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)(3,12)(4,6)(5,15)(7,11)(8,16)(9,14)(10,13)(17,18,24,27)(19,25,22,23)(20,30,21,26)(28,31,29,32)(34,40)(35,44)(36,46)(43,48);(1,3)(2,8)(4,13)(5,14)(6,15)(7,16)(9,10)(11,12)(17,19,24,22)(18,25,27,23)(20,29,21,28)(26,31,30,32)(33,34)(35,41)(36,47)(37,43)(38,44)(39,40)(42,46)(45,48);(17,20,24,21)(18,26,27,30)(19,28,22,29)(23,31,25,32)(33,35,39,44)(34,41,40,38)(36,45,46,37)(42,43,47,48);(1,4,7,14)(2,9,11,6)(3,13,16,5)(8,10,12,15)(17,21,24,20)(18,26,27,30)(19,29,22,28)(23,31,25,32)(33,36)(34,42)(35,45)(37,44)(38,48)(39,46)(40,47)(41,43);(1,5,7,13)(2,10,11,15)(3,14,16,4)(6,8,9,12)(17,22,24,19)(18,23,27,25)(20,29,21,28)(26,31,30,32)(33,37,39,45)(34,43,40,48)(35,36,44,46)(38,42,41,47);(1,6,7,9)(2,4,11,14)(3,15,16,10)(5,8,13,12)(17,23)(18,19)(20,31)(21,32)(22,27)(24,25)(26,28)(29,30)(33,38,39,41)(34,44,40,35)(36,43,46,48)(37,42,45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=gh,e^2=gi,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, 8353149);