#227145

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈²
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	#2358
prime radical	#3482	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=deh,e^b=eh,f^c=fi,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, 134341);