#8687359

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

3869 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,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, 8594555);