#7830831

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

5038 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)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,22)(19,26,20,24)(21,29,30,25)(27,31,28,32);(1,2)(3,7)(4,8)(5,6)(17,19,23,20)(18,24,22,26)(21,27,30,28)(25,31,29,32);(1,3,5,4)(2,7,6,8)(9,10,12,14)(11,13,16,15)(18,22)(19,20)(25,29)(27,28);(1,3,5,4)(2,7,6,8)(9,10)(11,13)(12,14)(15,16)(17,20,23,19)(18,24,22,26)(21,27,30,28)(25,32,29,31);(9,11)(10,13)(12,16)(14,15)(17,21,23,30)(18,25,22,29)(19,27,20,28)(24,31,26,32);(1,4,5,3)(2,8,6,7)(17,22,23,18)(19,24,20,26)(21,29,30,25)(27,31,28,32)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=g,e^2=i,f^2=g,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, 7738027);