#7672111

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	#59124	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6124 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=g,e^2=ghi,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, 7579307);