#8932200

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

3773 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=gh,e^2=gi,f^2=h,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, 8839396);