#7677953

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	#59034	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4679 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=h,e^2=i,f^2=g,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, 7585149);