#405920

order	512
nilpotency class	3
isoclinism	isoclinic class of #395586
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2411
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14921
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1156 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338488, #72338489, #72338500
maximal normal subgroup of	#72338488, #72338489, #72338500

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313116);