#7678726

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	#45079	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4884 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=ghi,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, 7585922);