#10155649

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

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

5632 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gi,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262868, #72262931, #72262970
maximal normal subgroup of	#72262868, #72262931, #72262970

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10062845);