#8925025

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4376 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,e^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, 8832221);