#448774

order	512
nilpotency class	3
isoclinism	isoclinic class of #428059
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

703 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,11,16)(7,17,21,33)(8,22,24,13)(9,25,23,14)(10,28,27,15)(12,29,31,20)(18,34,19,32)(26,30,35,36);(5,7,10,20,11,21,27,29)(6,12,15,17,16,31,28,33)(8,18,9,26,24,19,23,35)(13,30,25,34,22,36,14,32);(5,8,10,23,11,24,27,9)(6,13,15,14,16,22,28,25)(7,18,20,35,21,19,29,26)(12,30,17,32,31,36,33,34);(2,4)(5,9,27,24,11,23,10,8)(6,14,28,13,16,25,15,22)(7,19,20,26,21,18,29,35)(12,30,17,32,31,36,33,34)

Polycyclic presentation:

2^9|a^2=i,b^2=ef,c^2=e,d^2=eg,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,g^b=gi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 355970);