#8893712

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

4247 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8800908);