#220233

order	512
nilpotency class	3
isoclinism	isoclinic class of #25443
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	D₄×C₂²⋊C₄
prime radical	#60694	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

859 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ci,d^b=de,e^b=eh,d^c=di,f^c=fh,g^c=gh,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127429);