#7693002

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

10875 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:

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

Polycyclic presentation:

2^9|a^2=ghi,d^2=h,e^2=i,f^2=i,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, 7600198);