#7670768

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	#35000	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5448 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7577964);