#118351

order	512
nilpotency class	3
isoclinism	isoclinic class of #118350
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁶	Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1018 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,e^b=eh,f^b=fi,d^c=di,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235809, #72238983
maximal normal subgroup of	#72235809, #72238983

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25547);