#7671276

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	#59130	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5127 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=g,d^2=h,e^2=gh,f^2=gh,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, 7578472);