#407059

order	512
nilpotency class	3
isoclinism	isoclinic class of #405373
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2853
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14946
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1182 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338488, #72338489, #72338495
maximal normal subgroup of	#72338488, #72338489, #72338495

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 314255);