#8893630

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	#57182	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4081 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)(17,18,23,25)(19,29,32,24)(20,31,30,26)(21,27,28,22)(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)(17,19,23,32)(18,24,25,29)(20,28,30,21)(22,31,27,26)(33,35)(34,39)(36,43)(37,44)(38,42)(40,47)(41,46)(45,48);(5,13)(8,10)(11,16)(14,15)(18,25)(22,27)(24,29)(26,31)(35,42)(39,46)(43,47)(44,48);(1,3,6,2)(4,7,12,9)(5,10,13,8)(11,14,16,15)(17,20)(18,26)(19,28)(21,32)(22,24)(23,30)(25,31)(27,29)(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)(17,21,23,28)(18,27,25,22)(19,30,32,20)(24,31,29,26)(33,37)(34,40)(35,44)(36,41)(38,45)(39,47)(42,48)(43,46);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22,23,27)(18,28,25,21)(19,31,32,26)(20,24,30,29)(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=g,d^2=gh,e^2=g,f^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, 8800826);