#407033

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	#14945
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1076 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)(3,13)(4,16)(5,9)(6,11)(7,12)(8,15)(10,22)(14,17)(18,26)(19,27)(20,25)(21,24)(23,29)(28,31)(30,32);(1,3,6,15,7,16,22,26)(2,8,10,13,12,18,11,4)(5,14,21,30,20,31,19,29)(9,23,27,28,25,32,24,17);(1,4,6,18,7,13,22,8)(2,3,10,26,12,16,11,15)(5,17,21,32,20,28,19,23)(9,14,27,29,25,31,24,30);(1,5)(2,9)(3,14)(4,17)(6,21)(7,20)(8,23)(10,27)(11,24)(12,25)(13,28)(15,30)(16,31)(18,32)(19,22)(26,29)

Polycyclic presentation:

2^9|b^2=efh,c^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,h^b=hi,e^c=ei,f^c=fi,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	#72319325, #72319399, #72319460
maximal normal subgroup of	#72319325, #72319399, #72319460

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 314229);