#227191

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2358
prime radical	#2006	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

670 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=ci,d^b=dei,e^b=eh,d^c=di,f^c=fh,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, 134387);