#9862981

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	#54894	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4022 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)(3,8)(4,10)(5,11)(6,13)(7,9)(12,16)(14,15)(17,18,22,24)(19,26,30,23)(20,29,31,25)(21,28,32,27)(33,34,39,41)(35,40,38,48)(36,42,45,37)(43,44,46,47);(1,3,7,4)(2,8,9,10)(5,13,15,12)(6,14,16,11)(17,19)(18,23)(20,28)(21,29)(22,30)(24,26)(25,32)(27,31)(33,35,39,38)(34,40,41,48)(36,43,45,46)(37,47,42,44);(2,9)(3,4)(11,14)(12,13)(18,24)(20,31)(23,26)(27,28)(34,41)(36,45)(40,48)(43,46);(1,4,7,3)(2,10,9,8)(5,12,15,13)(6,11,16,14)(17,20)(18,25)(19,28)(21,26)(22,31)(23,32)(24,29)(27,30)(33,36)(34,42)(35,43)(37,41)(38,46)(39,45)(40,44)(47,48);(1,5)(2,11)(3,13)(4,12)(6,8)(7,15)(9,14)(10,16)(17,19)(18,26)(20,28)(21,25)(22,30)(23,24)(27,31)(29,32)(33,37,39,42)(34,36,41,45)(35,44,38,47)(40,46,48,43);(1,6,7,16)(2,12,9,13)(3,14,4,11)(5,10,15,8)(17,21)(18,27)(19,29)(20,26)(22,32)(23,31)(24,28)(25,30)(33,38,39,35)(34,40,41,48)(36,46,45,43)(37,47,42,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=i,e^2=ghi,f^2=gh,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, 9770177);