#8695855

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	#51137	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4222 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=gh,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, 8603051);