#462296

order	512
nilpotency class	3
isoclinism	isoclinic class of #458126
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

674 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)(5,6,12,16,11,17,10,27)(7,18,26,30,20,15,9,13)(8,21,23,31,24,14,28,33)(19,32,35,29,22,34,25,36);(5,7,10,9,11,20,12,26)(6,13,16,18,17,30,27,15)(8,22,28,35,24,19,23,25)(14,29,33,34,21,36,31,32);(5,8,10,28,11,24,12,23)(6,14,16,33,17,21,27,31)(7,19,9,25,20,22,26,35)(13,29,18,34,30,36,15,32);(2,4)(5,9,10,20,11,26,12,7)(6,15,16,13,17,18,27,30)(8,22,23,25,24,19,28,35)(14,29,31,32,21,36,33,34)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 369492);