#7681337

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	#45079	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4886 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)(3,4)(5,8)(6,7)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,21)(22,24)(23,27)(26,31)(28,29)(30,32);(1,3,6,5)(2,4,7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,30)(21,29)(22,27)(24,25)(26,28)(31,32);(2,7)(4,8)(18,24)(19,27)(20,26)(29,32);(1,4,6,8)(2,3,7,5)(9,11,12,15)(10,13,14,16)(17,19)(18,25)(20,28)(21,29)(22,27)(23,24)(26,30)(31,32);(1,3,6,5)(2,4,7,8)(17,20,22,26)(18,21,24,31)(19,28,27,30)(23,32,25,29);(1,5,6,3)(2,8,7,4)(17,21)(18,26)(19,29)(20,24)(22,31)(23,28)(25,30)(27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7588533);