#230382

order	512
nilpotency class	3
isoclinism	isoclinic class of #20022
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

526 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321848, #72322038, #72322056
maximal normal subgroup of	#72321848, #72322038, #72322056

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137578);