#8496532

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	#45604	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4099 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=g,e^2=g,f^2=gi,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, 8403728);