#227192

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

653 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^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, 134388);