#8492947

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

3954 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)(17,18,23,27)(19,24,28,32)(20,22,26,31)(21,30,29,25)(33,34)(35,43)(36,47)(37,48)(38,42)(39,40)(41,44)(45,46);(2,8)(5,11)(9,14)(12,15)(19,28)(21,29)(24,32)(25,30)(34,40)(35,44)(36,48)(42,46);(1,3,7,4)(2,9,8,14)(5,12,11,15)(6,13,10,16)(17,19,23,28)(18,24,27,32)(20,29,26,21)(22,25,31,30)(33,35,39,44)(34,41,40,43)(36,45,48,38)(37,46,47,42);(1,4,7,3)(2,9,8,14)(5,15,11,12)(6,13,10,16)(17,20)(18,22)(19,29)(21,28)(23,26)(24,25)(27,31)(30,32)(33,36)(34,37)(35,45)(38,44)(39,48)(40,47)(41,46)(42,43);(1,5,7,11)(2,10,8,6)(3,12,4,15)(9,16,14,13)(17,21,23,29)(18,25,27,30)(19,20,28,26)(22,24,31,32)(33,37)(34,36)(35,46)(38,43)(39,47)(40,48)(41,45)(42,44);(1,6,7,10)(2,11,8,5)(3,13,4,16)(9,15,14,12)(17,22)(18,26)(19,25)(20,27)(21,24)(23,31)(28,30)(29,32)(33,38,39,45)(34,42,40,46)(35,36,44,48)(37,43,47,41)

Polycyclic presentation:

2^9|a^2=gi,c^2=ghi,d^2=gh,e^2=gi,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, 8400143);