#10325614

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	#57970	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4171 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)(3,8)(4,10)(5,11)(6,13)(7,9)(12,16)(14,15)(17,18)(19,25)(20,28)(21,23)(22,27)(24,30)(26,31)(29,32)(33,34,38,42)(35,39,43,48)(36,40,46,45)(37,41,47,44);(1,3,7,5)(2,8,9,11)(4,13,14,12)(6,15,16,10)(17,19,21,27)(18,22,23,25)(20,29,30,26)(24,32,28,31)(35,43)(37,47)(39,48)(41,44);(2,9)(3,5)(10,15)(12,13)(18,23)(22,25)(24,28)(31,32)(33,35)(34,39)(36,41)(37,45)(38,43)(40,47)(42,48)(44,46);(1,4)(2,10)(3,12)(5,13)(6,11)(7,14)(8,16)(9,15)(17,20,21,30)(18,24,23,28)(19,26,27,29)(22,31,25,32)(33,35)(34,39)(36,41)(37,45)(38,43)(40,47)(42,48)(44,46);(1,5,7,3)(2,11,9,8)(4,13,14,12)(6,15,16,10)(17,18)(19,22)(20,24)(21,23)(25,27)(26,31)(28,30)(29,32)(33,36)(34,40)(35,44)(37,39)(38,46)(41,43)(42,45)(47,48);(1,6)(2,12)(3,10)(4,11)(5,15)(7,16)(8,14)(9,13)(17,19)(18,22)(20,26)(21,27)(23,25)(24,31)(28,32)(29,30)(33,37)(34,41)(35,45)(36,39)(38,47)(40,43)(42,44)(46,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=g,e^2=gh,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, 10232810);