#7673643

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	#44165	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4628 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7580839);