#118352

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	C₂⁵	C₂×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

750 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,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
maximal normal subgroup of	#72235809

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25548);