#493445

order	512
nilpotency class	4
isoclinism	isoclinic class of #480315
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	C₂×#1782
prime radical	#62815	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ #117 ◁ #3428 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2217 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)(3,13,15,9)(4,18,17,27)(5,10,21,22)(6,11)(7,12)(8,16)(14,26)(19,32)(20,25,28,31)(23,29)(24,30);(1,3)(2,8)(4,14)(5,16)(6,17)(7,15)(9,22)(10,13)(11,24)(12,23)(18,31)(19,28)(20,30)(21,29)(25,27)(26,32);(1,4,7,17)(2,9)(3,14,15,6)(5,19,21,30)(8,22)(10,23)(11,18)(12,13)(16,28,29,20)(24,31)(25,32)(26,27);(3,15)(4,17)(8,23)(9,13)(16,29)(18,27)(19,30)(24,32)

Polycyclic presentation:

2^9|a^2=g,c^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,g^b=gh,d^c=di,g^c=gi,g^e=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261390, #72261408, #72261480
maximal normal subgroup of	#72261390, #72261408, #72261480

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400641);