#7631117

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	#36126	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5480 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 24 points:

(2,4)(6,7)(10,15)(11,16)(18,21)(22,24);(1,2,5,4)(3,7,8,6)(9,10)(11,12)(13,16)(14,15)(17,18)(19,22)(20,21)(23,24);(1,3)(2,6)(4,7)(5,8)(9,11,14,16)(10,12,15,13);(3,8)(6,7)(9,12,14,13)(10,11,15,16);(1,3)(2,6)(4,7)(5,8)(9,12,14,13)(10,11,15,16);(1,4,5,2)(3,7,8,6)(9,13,14,12)(10,16,15,11)(17,19,20,23)(18,22,21,24)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=h,e^2=h,f^2=g,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, 7538313);