#7707054

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5446 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)(3,6)(4,5)(7,8)(9,10,13,15)(11,14,12,16)(19,20)(22,24);(2,5)(6,8)(17,18,21,23)(19,24,20,22);(9,10)(11,16)(12,14)(13,15)(17,19)(18,22)(20,21)(23,24);(9,11,13,12)(10,14,15,16)(18,23)(22,24);(9,12,13,11)(10,14,15,16)(17,20,21,19)(18,22,23,24);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304030
maximal normal subgroup of	#72304030
maximal quotient of	#72304050

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7614250);