#7782088

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	#37302	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4910 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)(9,10,13,12)(11,15,16,14)(17,18,23,27)(19,28,29,24)(20,21,25,31)(22,26,30,32);(1,2,5,4)(3,6,8,7)(9,11,13,16)(10,14,12,15)(17,19,23,29)(18,24,27,28)(20,30,25,22)(21,26,31,32);(2,4)(6,7)(9,12,13,10)(11,15,16,14)(19,29)(22,30)(24,28)(26,32);(1,2,5,4)(3,7,8,6)(17,20)(18,21)(19,30)(22,29)(23,25)(24,26)(27,31)(28,32);(1,3,5,8)(2,6,4,7)(17,21)(18,25)(19,26)(20,27)(22,24)(23,31)(28,30)(29,32);(1,4,5,2)(3,7,8,6)(17,22,23,30)(18,26,27,32)(19,20,29,25)(21,24,31,28)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7689284);