#229580

order	512
nilpotency class	3
isoclinism	isoclinic class of #20286
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	#2361
prime radical	#3106	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

532 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319585, #72322038, #72322139
maximal normal subgroup of	#72319585, #72322038, #72322139

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 136776);