#7690962

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

11254 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)(9,10,14,17)(11,15,19,12)(13,16,22,24)(18,21,23,20);(5,6)(7,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,21)(20,24)(22,23);(2,4)(9,11,14,19)(10,12,17,15)(13,18,22,23)(16,20,24,21);(1,2)(3,4)(9,11)(10,15)(12,17)(13,18)(14,19)(16,21)(20,24)(22,23);(9,12,14,15)(10,11,17,19)(13,21,22,20)(16,23,24,18);(9,13,14,22)(10,16,17,24)(11,18,19,23)(12,20,15,21)

Polycyclic presentation:

2^9|a^2=gi,c^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, #72304577, #72304611
maximal normal subgroup of	#72284809, #72304577, #72304611
maximal quotient of	#72267279, #72267303, #72267309

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7598158);