#7689780

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	#34655	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,e^2=ghi,f^2=i,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7596976);