#7690200

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	#34727	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4967 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=h,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, 7597396);