#272024

order	512
nilpotency class	3
isoclinism	isoclinic class of #271963
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#537	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

800 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)(3,12)(4,9)(5,14)(6,13)(7,10)(8,11)(15,16)(17,18,23,29,22,20,21,25)(19,24,30,32,27,26,28,31);(1,3,8,15)(2,5,11,4)(6,9,7,14)(10,16,13,12)(17,19)(18,24)(20,26)(21,28)(22,27)(23,30)(25,31)(29,32);(1,4,8,5)(2,9,11,14)(3,13,15,10)(6,16,7,12)(17,20,22,18)(19,26,27,24)(21,25,23,29)(28,31,30,32);(1,5,8,4)(2,3,11,15)(6,16,7,12)(9,13,14,10)(19,27)(24,26)(28,30)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 179220);