#8752256

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	#56402	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4008 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8659452);