#8425424

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3888 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,d^2=h,e^2=g,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, 8332620);