#8434152

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	#57477	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4154 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8341348);