#272419

order	512
nilpotency class	3
isoclinism	isoclinic class of #272411
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	#3481	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

840 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)(3,11)(4,8)(5,12)(6,9)(7,10)(13,15)(14,16)(17,18,21,25,22,19,26,24)(20,28,30,27,31,32,29,23);(1,3)(2,4)(5,13)(6,8)(7,14)(9,16)(10,15)(11,12)(17,19)(18,22)(20,27)(21,24)(23,31)(25,26)(28,29)(30,32);(1,4)(2,8)(3,12)(5,11)(6,16)(7,15)(9,14)(10,13)(17,20,22,31)(18,23,19,27)(21,29,26,30)(24,28,25,32);(1,4)(2,3)(5,16)(6,11)(7,15)(8,12)(9,13)(10,14)(17,19)(18,22)(20,27)(21,24)(23,31)(25,26)(28,29)(30,32)

Polycyclic presentation:

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

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, 179615);