#8925015

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	#47026	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4114 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8832211);