#7631145

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

5210 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538341);