#8745958

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3731 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)(3,10,5,8)(4,9,15,14)(6,11,16,13)(17,18,23,26)(19,20,22,25)(21,24,30,32)(27,28,29,31)(33,34,39,38)(35,42,37,40)(36,41,46,45)(43,48,44,47);(1,3,7,5)(2,8,12,10)(4,11,15,13)(6,14,16,9)(17,19,23,22)(18,20,26,25)(21,29,30,27)(24,31,32,28)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(1,3,7,5)(2,8,12,10)(4,11,15,13)(6,14,16,9)(17,20,23,25)(18,19,26,22)(21,28,30,31)(24,27,32,29)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,4,7,15)(2,9,12,14)(3,13,5,11)(6,8,16,10)(17,21,23,30)(18,24,26,32)(19,27,22,29)(20,28,25,31)(33,37,39,35)(34,42,38,40)(36,44,46,43)(41,48,45,47);(1,5,7,3)(2,10,12,8)(4,11,15,13)(6,14,16,9)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32)(33,34,39,38)(35,40,37,42)(36,41,46,45)(43,47,44,48);(1,6)(2,11)(3,14)(4,10)(5,9)(7,16)(8,15)(12,13)(17,22,23,19)(18,25,26,20)(21,29,30,27)(24,31,32,28)(33,38,39,34)(35,42,37,40)(36,45,46,41)(43,48,44,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=hi,d^2=hi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8653154);