#8095387

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	#45609	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4054 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,d^2=gh,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002583);