#111148

order	512
nilpotency class	3
isoclinism	isoclinic class of #9183
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	#1437
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

792 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,f^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=dh,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239054
maximal normal subgroup of	#72239054
maximal quotient of	#72239114, #72239116

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18344);