#8479558

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

3854 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=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, 8386754);