#7690174

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4624 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7597370);