#242965

order	512
nilpotency class	3
isoclinism	isoclinic class of #242962
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

648 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:

(1,2,5,7,6,10,4,9)(3,11,13,16,12,15,14,8);(2,7)(3,12)(4,5)(8,15)(9,10)(11,16)(17,18,21,24)(19,25,26,32)(20,28,30,23)(22,29,31,27);(1,3,4,13,6,12,5,14)(2,8,9,11,10,16,7,15)(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=f,b^2=i,c^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,g^a=gh,c^b=cg,e^b=eh,f^b=fh,g^b=gi,d^c=di,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238937, #72238938, #72238948
maximal normal subgroup of	#72238937, #72238938, #72238948

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 150161);