#128194

order	512
nilpotency class	4
isoclinism	isoclinic class of #128055
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#342
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

793 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gi,f^2=g,b^a=bd,d^a=dg,f^a=fh,c^b=ch,d^b=dh,e^b=eg,h^b=hi,f^c=fi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238503, #72238506, #72238509, …
maximal normal subgroup of	#72238503, #72238506, #72238509, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35390);