#221265

order	512
nilpotency class	3
isoclinism	isoclinic class of #221184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61215	C₂
LCS	G ▷ 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:

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

Polycyclic presentation:

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

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, 128461);