#225725

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈²
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂×#135	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

565 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 132921);