#7679880

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	#39126	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4624 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=g,e^2=gh,f^2=i,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, 7587076);