#8461740

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	#54997	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4088 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)(3,9,4,6)(5,13,15,10)(11,14,12,16)(17,18)(19,23)(20,25)(21,26)(22,27)(24,29)(28,30)(31,32)(36,44)(41,45)(42,48)(46,47);(2,8)(3,4)(10,13)(11,12)(17,19,22,29)(18,23,27,24)(20,26,30,32)(21,28,31,25)(33,34,38,37)(35,40,43,39)(36,41,44,45)(42,47,48,46);(1,3,7,4)(2,6,8,9)(5,11,15,12)(10,14,13,16)(17,19)(18,24)(20,26)(21,28)(22,29)(23,27)(25,31)(30,32)(33,35)(34,39)(36,42)(37,40)(38,43)(41,46)(44,48)(45,47);(1,4,7,3)(2,9,8,6)(5,12,15,11)(10,16,13,14)(17,19)(18,23)(20,26)(21,25)(22,29)(24,27)(28,31)(30,32)(33,35)(34,40)(36,42)(37,39)(38,43)(41,47)(44,48)(45,46);(1,5,7,15)(2,10,8,13)(3,11,4,12)(6,14,9,16)(17,20,22,30)(18,25,27,28)(19,26,29,32)(21,24,31,23)(33,36,38,44)(34,41,37,45)(35,42,43,48)(39,46,40,47);(1,6,7,9)(2,3,8,4)(5,14,15,16)(10,11,13,12)(17,21)(18,26)(19,28)(20,24)(22,31)(23,30)(25,29)(27,32)(33,37,38,34)(35,40,43,39)(36,45,44,41)(42,47,48,46)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=g,e^2=i,f^2=ghi,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, 8368936);