#8455700

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3898 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=hi,d^2=gh,e^2=ghi,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, 8362896);