#227582

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	#3481	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

657 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,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	#72287239, #72322005, #72322077
maximal normal subgroup of	#72287239, #72322005, #72322077

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 134778);