#239927

order	512
nilpotency class	3
isoclinism	isoclinic class of #239921
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1040 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=cg,e^b=eh,d^c=di,f^c=fhi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238937, #72238938, #72238959
maximal normal subgroup of	#72238937, #72238938, #72238959

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 147123);