#228251

order	512
nilpotency class	3
isoclinism	isoclinic class of #20022
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	#3475	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

661 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 135447);