#8475830

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	#57509	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4064 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8383026);