#8445688

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	#2699	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3968 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:

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

Polycyclic presentation:

2^9|a^2=gi,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, 8352884);