#7838522

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	#44398	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4608 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7745718);