#480813

order	512
nilpotency class	4
isoclinism	isoclinic class of #480674
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂×#532	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

932 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,d^2=hi,e^2=i,f^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=chi,e^b=ei,f^b=fgi,f^e=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308577, #72308672
maximal normal subgroup of	#72308577, #72308672

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388009);