#7688123

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

10813 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)(9,10,14,11)(12,15,20,17)(13,16,22,18)(19,23,21,24);(5,6)(7,8)(9,10,14,11)(12,15,20,17)(13,16,22,18)(19,23,21,24);(2,3)(10,11)(15,17)(16,18)(23,24);(1,3,4,2)(9,11,14,10)(12,17,20,15)(13,18,22,16)(19,24,21,23);(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23);(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,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, 7595319);