#7631712

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	#36360	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5112 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538908);