#482429

order	512
nilpotency class	4
isoclinism	isoclinic class of #482362
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

942 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)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18)(19,25)(20,28,29,22)(21,24)(23,26)(27,30,32,31);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(18,22,24,28)(19,26)(20,29)(23,31,25,30);(2,7)(6,14)(8,15)(9,16)(17,19,21,26)(18,23,24,25)(20,27,29,32)(22,30,28,31);(17,20)(18,22)(19,27)(21,29)(23,30)(24,28)(25,31)(26,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,e^2=hi,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=di,e^b=eh,f^b=fghi,f^d=fi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308008, #72308012, #72308042
maximal normal subgroup of	#72308008, #72308012, #72308042

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389625);