#409949

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1876 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307883, #72307918, #72307954
maximal normal subgroup of	#72307883, #72307918, #72307954

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317145);