#331105

order	512
nilpotency class	3
isoclinism	isoclinic class of #331095
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#33171	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

914 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2)(3,6)(4,5)(7,8)(9,10,13,16,12,14,11,15)(17,18,22,19,21,24,20,23);(2,5)(6,8)(10,14)(15,16)(17,19,22,24,21,23,20,18);(1,3)(2,6)(4,7)(5,8)(10,15)(11,13)(14,16)(18,19)(20,22)(23,24);(3,7)(6,8)(9,11,12,13)(10,15,14,16)(17,20,21,22)(18,23,24,19)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238301);