#10332181

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	#38014	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4056 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:

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

Polycyclic presentation:

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

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, 10239377);