#7687748

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	#59022	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4777 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=ghi,e^2=g,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, 7594944);