#300252

order	512
nilpotency class	3
isoclinism	isoclinic class of #300205
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

909 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fh,d^2=ehi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=dh,g^b=gi,d^c=dh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207448);