#7670704

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	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4885 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7577900);