#227586

order	512
nilpotency class	3
isoclinism	isoclinic class of #19755
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	#18480	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

679 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287562, #72322005, #72322038, …
maximal normal subgroup of	#72287562, #72322005, #72322038, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 134782);