#7846759

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

4971 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=gi,f^2=gi,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, 7753955);