#8927341

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4152 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:

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

Polycyclic presentation:

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

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, 8834537);