#8932158

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=g,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, 8839354);