#8430632

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4099 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)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,18,24,27)(19,26,22,25)(20,23,21,31)(28,32,29,30)(36,44)(40,47)(42,48)(45,46);(3,10)(6,7)(11,16)(13,14)(17,19,24,22)(18,25,27,26)(20,28,21,29)(23,30,31,32)(33,34)(35,39)(36,40)(37,41)(38,43)(42,46)(44,47)(45,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,20,24,21)(18,23,27,31)(19,28,22,29)(25,30,26,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,21,24,20)(18,23,27,31)(19,29,22,28)(25,30,26,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,4,5,12)(2,8,9,15)(3,11,10,16)(6,13,7,14)(17,22,24,19)(18,26,27,25)(20,29,21,28)(23,32,31,30)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48);(1,2)(3,6)(4,8)(5,9)(7,10)(11,13)(12,15)(14,16)(17,23)(18,21)(19,30)(20,27)(22,32)(24,31)(25,29)(26,28)(33,34)(35,38)(36,40)(37,41)(39,43)(42,45)(44,47)(46,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8337828);