#331021

order	512
nilpotency class	3
isoclinism	isoclinic class of #331007
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3205	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

501 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)(3,6,11,15)(4,7,12,16)(9,13,10,14)(17,18,22,20,23,26,21,25)(19,27,29,24,30,31,28,32);(1,3)(2,6)(4,9)(5,11)(7,13)(8,15)(10,12)(14,16)(17,19)(18,24)(20,27)(21,29)(22,28)(23,30)(25,31)(26,32);(3,9)(4,12)(6,13)(7,16)(10,11)(14,15)(18,20)(19,28)(21,22)(25,26)(27,31)(29,30);(17,20,21,18,23,25,22,26)(19,27,29,24,30,31,28,32)

Polycyclic presentation:

2^9|a^2=ei,d^2=f,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,g^a=gh,c^b=cg,e^b=eh,g^b=gi,d^c=de,e^c=eh,f^c=fh,g^c=gi,g^d=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238217);