#7672313

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	#59755	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4724 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)(5,8)(6,7)(9,10,12,14)(11,13,15,16)(17,18,23,20)(19,27,29,24)(21,25,31,28)(22,30,32,26);(1,3,6,5)(2,4,7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,24)(20,27)(21,26)(22,28)(23,29)(25,32)(30,31);(2,7)(4,8)(17,18,23,20)(19,27,29,24)(21,28,31,25)(22,26,32,30);(1,4,6,8)(2,3,7,5)(9,11,12,15)(10,13,14,16)(17,20,23,18)(19,27,29,24)(21,28,31,25)(22,30,32,26);(1,3,6,5)(2,4,7,8)(17,21,23,31)(18,25,20,28)(19,26,29,30)(22,24,32,27);(1,5,6,3)(2,8,7,4)(17,22)(18,26)(19,28)(20,30)(21,24)(23,32)(25,29)(27,31)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7579509);