#10407745

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	#56397	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4113 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=ghi,e^2=hi,f^2=hi,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, 10314941);