#300156

order	512
nilpotency class	3
isoclinism	isoclinic class of #300141
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	#19292	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

958 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=fh,d^2=eh,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,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207352);