#8445687

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3937 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)(3,13)(4,10)(5,7)(6,16)(8,12)(9,15)(11,14)(17,18,23,27)(19,22,26,31)(20,30,29,24)(21,25,28,32)(34,39)(35,42)(36,44)(46,47);(1,3,8,15)(2,9,12,13)(4,14,5,6)(7,11,10,16)(17,19,23,26)(18,22,27,31)(20,21,29,28)(24,32,30,25)(33,34,38,39)(35,40,42,37)(36,45,44,41)(43,47,48,46);(1,4,8,5)(2,10,12,7)(3,14,15,6)(9,16,13,11)(17,20,23,29)(18,24,27,30)(19,28,26,21)(22,25,31,32)(33,35)(34,40)(36,43)(37,39)(38,42)(41,46)(44,48)(45,47);(1,5,8,4)(2,10,12,7)(3,6,15,14)(9,16,13,11)(17,19)(18,22)(20,28)(21,29)(23,26)(24,25)(27,31)(30,32)(33,36,38,44)(34,41,39,45)(35,43,42,48)(37,47,40,46);(1,6)(2,11)(3,4)(5,15)(7,13)(8,14)(9,10)(12,16)(17,21,23,28)(18,25,27,32)(19,29,26,20)(22,24,31,30);(1,7,8,10)(2,5,12,4)(3,11,15,16)(6,13,14,9)(17,22)(18,26)(19,27)(20,25)(21,24)(23,31)(28,30)(29,32)(33,37,38,40)(34,35,39,42)(36,46,44,47)(41,48,45,43)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8352883);