#449053

order	512
nilpotency class	3
isoclinism	isoclinic class of #421447
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2474
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15929
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

815 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)(3,12,17,14)(4,18)(5,10)(6,22)(7,11)(8,15,25,16)(9,20)(13,24,19,26)(21,27)(23,28)(29,32,30,31);(1,3,6,16,7,17,5,15)(2,8,10,12,11,25,22,14)(4,19,20,32,21,13,23,31)(9,26,18,29,28,24,27,30);(1,4)(2,9)(3,13)(5,20)(6,23)(7,21)(8,24)(10,27)(11,28)(12,29)(14,30)(15,31)(16,32)(17,19)(18,22)(25,26);(1,2)(3,14,17,12)(4,9)(5,10)(6,22)(7,11)(8,16,25,15)(13,30,19,29)(18,23)(20,27)(21,28)(24,32,26,31)

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,g^b=gi,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319788
maximal normal subgroup of	#72319788

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356249);