#8755018

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3771 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

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, 8662214);