#7712334

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	#36258	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4930 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619530);