#7686115

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	#36152	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6614 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7593311);