#8497844

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	#53403	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3960 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=g,e^2=gh,f^2=hi,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, 8405040);