#7683824

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4524 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)(9,10)(11,15)(12,13)(14,16)(17,18,23,21)(19,27,28,24)(20,25,30,31)(22,29,32,26);(1,2,4,5)(3,6,7,8)(9,11,14,12)(10,15,16,13)(17,19)(18,24)(20,29)(21,27)(22,25)(23,28)(26,30)(31,32);(9,12,14,11)(10,15,16,13)(19,28)(22,32)(24,27)(26,29);(9,13,14,15)(10,12,16,11)(17,20)(18,25)(19,29)(21,31)(22,24)(23,30)(26,28)(27,32);(17,21,23,18)(19,24,28,27)(20,25,30,31)(22,29,32,26);(1,3,4,7)(2,6,5,8)(17,22,23,32)(18,26,21,29)(19,25,28,31)(20,24,30,27)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,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,f^c=fi,e^d=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, 7591020);