#7837028

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4646 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gh,d^2=gi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7744224);