#7682188

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

5400 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7589384);