#7678593

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4450 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=i,d^2=gh,e^2=i,f^2=h,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, 7585789);