#7897658

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	#39128	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4691 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7804854);