#8925056

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

4325 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)(3,11)(4,10)(5,8)(6,14)(7,9)(12,13)(15,16)(19,25)(23,30)(26,28)(31,32)(33,34,40,45)(35,46,48,41)(36,42,38,44)(37,43,47,39);(1,3,7,5)(2,8,9,11)(4,12,14,15)(6,13,10,16)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,35,40,48)(34,41,45,46)(36,47,38,37)(39,42,43,44);(2,9)(6,10)(8,11)(13,16)(18,22)(23,30)(24,27)(31,32)(33,36,40,38)(34,42,45,44)(35,47,48,37)(39,41,43,46);(1,4)(2,10)(3,12)(5,15)(6,9)(7,14)(8,16)(11,13)(17,19)(18,23)(20,28)(21,25)(22,30)(24,32)(26,29)(27,31)(33,37)(34,43)(35,36)(38,48)(39,45)(40,47)(41,44)(42,46);(1,5,7,3)(2,8,9,11)(4,12,14,15)(6,16,10,13)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,38,40,36)(34,44,45,42)(35,47,48,37)(39,41,43,46);(1,6,7,10)(2,4,9,14)(3,13,5,16)(8,12,11,15)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(33,39,40,43)(34,37,45,47)(35,42,48,44)(36,46,38,41)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,e^2=g,f^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, 8832252);