#348123

order	512
nilpotency class	3
isoclinism	isoclinic class of #348118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×H(5,𝔽₂)	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

965 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,c^b=ch,g^c=gi,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306466, #72306478, #72306558, …
maximal normal subgroup of	#72306466, #72306478, #72306558, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255319);