#7898053

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	C₂×#548	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4581 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7805249);