#8108007

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#34547	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5105 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

(3,7)(6,8)(10,12)(11,15)(17,18,24,28)(19,25,20,21)(22,27,23,26)(29,31,30,32);(1,2)(3,6)(4,5)(7,8)(9,10,13,12)(11,14,15,16)(17,19,24,20)(18,21,28,25)(22,29,23,30)(26,31,27,32);(2,5)(6,8)(11,15)(14,16)(17,20,24,19)(18,25,28,21)(22,30,23,29)(26,32,27,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,16)(13,15)(17,21,24,25)(18,20,28,19)(22,31,23,32)(26,30,27,29);(17,22,24,23)(18,26,28,27)(19,29,20,30)(21,31,25,32);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,16,15,14)(17,23,24,22)(18,27,28,26)(19,30,20,29)(21,32,25,31)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015203);