#7696346

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

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

5531 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7603542);