#249427

order	512
nilpotency class	3
isoclinism	isoclinic class of #249427
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1737 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,11)(4,15)(5,8)(6,10)(7,14)(9,19)(12,27)(13,28)(16,24)(17,21)(18,26)(20,25)(22,31)(23,30)(29,32);(4,16)(7,21)(9,26)(11,28)(14,31)(15,24)(17,30)(18,19)(22,23)(27,32);(1,3)(2,7)(4,17)(5,13)(6,12)(8,23)(9,27)(10,21)(11,15)(14,19)(16,30)(18,31)(20,29)(22,25)(24,32)(26,28);(2,8)(3,12)(4,18)(7,22)(10,25)(11,27)(13,29)(14,31)(16,19)(17,30)(21,23)(28,32)

Polycyclic presentation:

2^9|f^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,g^a=gi,c^b=cg,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283074, #72286165, #72287135
maximal normal subgroup of	#72283074, #72286165, #72287135
maximal quotient of	#72284659

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156623);