#7685732

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

11291 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,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.

Relationships

maximal subgroup of	#72284809, #72304587, #72304627
maximal normal subgroup of	#72284809, #72304587, #72304627

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7592928);