#8425627

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3960 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,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=dh,f^d=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, 8332823);