#8933102

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	#2812	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4090 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8840298);