#7720341

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	#37268	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4701 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)(10,13)(14,16)(17,18,23,20)(19,27,28,24)(21,25,31,30)(22,29,32,26);(1,3)(2,6)(4,7)(5,8)(9,10,12,13)(11,14,15,16)(17,19,23,28)(18,24,20,27)(21,26,31,29)(22,25,32,30);(2,4)(6,7)(19,28)(21,31)(24,27)(25,30);(1,4,5,2)(3,7,8,6)(9,11,12,15)(10,14,13,16)(17,20,23,18)(19,27,28,24)(21,30,31,25)(22,29,32,26);(1,3)(2,6)(4,7)(5,8)(17,21)(18,25)(19,29)(20,30)(22,24)(23,31)(26,28)(27,32);(17,22,23,32)(18,26,20,29)(19,25,28,30)(21,24,31,27)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7627537);