#8445703

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	#43328	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4022 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)(3,11)(4,10)(5,7)(6,9)(8,12)(13,15)(14,16)(17,18,23,27)(19,22,26,31)(20,30,29,24)(21,25,28,32)(34,40)(35,43)(36,45)(42,47);(1,3,8,6)(2,9,12,11)(4,13,5,14)(7,15,10,16)(17,19,23,26)(18,22,27,31)(20,21,29,28)(24,32,30,25)(33,34,39,40)(35,38,43,48)(36,46,45,41)(37,42,44,47);(1,4,8,5)(2,10,12,7)(3,13,6,14)(9,16,11,15)(17,20,23,29)(18,24,27,30)(19,28,26,21)(22,25,31,32)(33,35)(34,38)(36,44)(37,45)(39,43)(40,48)(41,42)(46,47);(1,5,8,4)(2,10,12,7)(3,14,6,13)(9,16,11,15)(17,19)(18,22)(20,28)(21,29)(23,26)(24,25)(27,31)(30,32)(33,36,39,45)(34,41,40,46)(35,44,43,37)(38,42,48,47);(1,6,8,3)(2,11,12,9)(4,14,5,13)(7,16,10,15)(17,21,23,28)(18,25,27,32)(19,29,26,20)(22,24,31,30)(33,37,39,44)(34,42,40,47)(35,45,43,36)(38,41,48,46);(1,7,8,10)(2,5,12,4)(3,15,6,16)(9,14,11,13)(17,22)(18,26)(19,27)(20,25)(21,24)(23,31)(28,30)(29,32)(33,38,39,48)(34,43,40,35)(36,47,45,42)(37,41,44,46)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=g,e^2=gi,f^2=g,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, 8352899);