#395558

order	512
nilpotency class	3
isoclinism	isoclinic class of #395558
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14725
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3027	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2203 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)(4,16,18,9)(5,20)(6,10)(7,11)(8,14)(13,27,29,26)(15,23)(17,25,19,31)(21,24)(22,28,30,32);(1,3,5,14,7,15,6,21)(2,8,10,12,11,24,20,23)(4,13,17,28,18,29,19,32)(9,22,25,26,16,30,31,27);(1,2)(3,12)(4,9)(5,10)(6,20)(7,11)(8,21)(13,26)(14,24)(15,23)(16,18)(17,25)(19,31)(22,32)(27,29)(28,30);(1,4)(2,9)(3,13)(5,17)(6,19)(7,18)(8,22)(10,25)(11,16)(12,26)(14,28)(15,29)(20,31)(21,32)(23,27)(24,30)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285906, #72286235, #72286253, …
maximal normal subgroup of	#72285906, #72286235, #72286253, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302754);