#8097089

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	#2828	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3949 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:

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

Polycyclic presentation:

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

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, 8004285);