#7630829

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	#34643	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4826 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^d=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, 7538025);