#8455736

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

3929 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)(17,18,22,26)(19,21,25,23)(20,24,29,32)(27,28,31,30)(33,34,38,37)(35,41,42,39)(36,40,44,45)(43,47,48,46);(1,3,8,15)(2,9,12,13)(4,14,5,6)(7,11,10,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(35,42)(36,44)(39,41)(40,45);(1,4,8,5)(2,10,12,7)(3,14,15,6)(9,16,13,11)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,48)(45,47);(1,5,8,4)(2,10,12,7)(3,6,15,14)(9,16,13,11)(17,18)(19,21)(20,24)(22,26)(23,25)(27,28)(29,32)(30,31)(33,36,38,44)(34,40,37,45)(35,43,42,48)(39,46,41,47);(1,6)(2,11)(3,4)(5,15)(7,13)(8,14)(9,10)(12,16)(17,20,22,29)(18,24,26,32)(19,27,25,31)(21,28,23,30);(1,7,8,10)(2,5,12,4)(3,11,15,16)(6,13,14,9)(17,21)(18,25)(19,26)(20,28)(22,23)(24,31)(27,32)(29,30)(33,37,38,34)(35,41,42,39)(36,40,44,45)(43,46,48,47)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=ghi,d^2=gh,e^2=hi,f^2=gh,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, 8362932);