#473199

order	512
nilpotency class	3
isoclinism	isoclinic class of #453980
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2782
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

572 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:

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

Polycyclic presentation:

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

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, 380395);