#8695971

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3979 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8603167);