#8925020

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	#46458	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)(17,18,23,20)(19,27,28,24)(21,25,32,29)(22,26,31,30)(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)(17,19,23,28)(18,24,20,27)(21,31,32,22)(25,26,29,30)(33,35)(34,40)(36,47)(37,44)(38,41)(39,45)(42,43)(46,48);(2,6)(7,14)(8,11)(15,16)(17,18,23,20)(19,24,28,27)(21,25,32,29)(22,30,31,26)(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)(17,20,23,18)(19,27,28,24)(21,25,32,29)(22,30,31,26)(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)(17,21)(18,25)(19,22)(20,29)(23,32)(24,30)(26,27)(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)(18,26)(19,21)(20,30)(23,31)(24,29)(25,27)(28,32)(33,38)(34,37)(35,41)(36,42)(39,48)(40,44)(43,47)(45,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,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, 8832216);