#7674501

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	#35274	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5530 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gh,d^2=hi,e^2=i,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, 7581697);