#94408

order	512
nilpotency class	5
isoclinism	isoclinic class of #94407
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7292
commutator	C₂×C₁₆	C₄²
Frattini, cosocle	C₄×M₃₂	C₂²
prime radical	C₂³	#365
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₄×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fi,d^2=g,e^2=fi,f^2=h,h^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fh,h^a=hi,c^b=cg,d^b=dh,g^b=gi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214531, #72214589, #72214592, …
maximal normal subgroup of	#72214531, #72214589, #72214592, …
maximal quotient of	#72216280

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1604);