#414608

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2024 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)(9,10,12,14)(11,15,16,13)(17,18)(19,27)(20,25)(21,24)(22,26)(23,29)(28,31)(30,32);(2,4)(5,6)(7,8)(17,19)(18,23)(20,30)(21,29)(22,28)(24,31)(25,27)(26,32);(6,8)(18,24)(20,21)(23,31)(25,26)(27,32)(29,30);(9,11,12,16)(10,13,14,15)(19,28)(23,32)(27,31)(29,30)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264176, #72264233, #72264342
maximal normal subgroup of	#72264176, #72264233, #72264342

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321804);