#8928035

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	#54835	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3852 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)(3,6)(4,13)(5,9)(7,10)(8,12)(11,16)(14,15)(17,18,23,28)(19,27,22,24)(20,21,26,25)(29,32,31,30)(34,39)(37,45)(40,46)(44,48);(1,3)(2,6)(4,5)(7,11)(8,14)(9,13)(10,16)(12,15)(17,19,23,22)(18,24,28,27)(20,31,26,29)(21,32,25,30)(36,43)(40,46)(42,47)(44,48);(3,11)(4,12)(6,16)(8,13)(17,20,23,26)(18,25,28,21)(19,29,22,31)(24,32,27,30)(33,34,38,39)(35,37,41,45)(36,40,43,46)(42,44,47,48);(1,4)(2,8)(3,5)(6,14)(7,12)(9,16)(10,13)(11,15)(17,18)(19,24)(20,25)(21,26)(22,27)(23,28)(29,32)(30,31)(33,35,38,41)(34,37,39,45)(36,42,43,47)(40,44,46,48);(1,5)(2,9)(3,12)(4,11)(6,8)(7,15)(10,14)(13,16)(17,21,23,25)(18,26,28,20)(19,30,22,32)(24,31,27,29)(33,36,38,43)(34,40,39,46)(35,42,41,47)(37,44,45,48);(1,6)(2,3)(4,14)(5,8)(7,16)(9,12)(10,11)(13,15)(17,22,23,19)(18,27,28,24)(20,31,26,29)(21,32,25,30)(33,37)(34,41)(35,39)(36,44)(38,45)(40,47)(42,46)(43,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=gh,e^2=h,f^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, 8835231);