#228247

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	#18522	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

690 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)(3,10)(4,9)(5,7)(6,13)(8,11)(12,16)(14,15)(17,18,23,26,24,28,22,27)(19,29,30,25,21,20,31,32);(2,7)(8,15)(9,13)(10,16)(17,19,23,30,24,21,22,31)(18,25,26,20,28,32,27,29);(1,3)(2,8)(4,11)(5,12)(6,14)(7,15)(9,16)(10,13)(17,20)(18,21)(19,28)(22,25)(23,32)(24,29)(26,31)(27,30);(1,4,5,6)(2,9,7,13)(3,11,12,14)(8,16,15,10)(17,21,24,19)(18,20,28,29)(22,30,23,31)(25,26,32,27)

Polycyclic presentation:

2^9|a^2=g,b^2=g,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, 135443);