#482282

order	512
nilpotency class	4
isoclinism	isoclinic class of #481619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#35692	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1139 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=hi,e^2=h,b^a=be,c^a=cgh,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,d^c=di,f^c=fi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234703, #72234738
maximal normal subgroup of	#72234703, #72234738

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389478);