#377680

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

929 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)(7,16)(8,12,21,28)(9,13)(10,15)(11,18)(14,22)(17,29,31,35)(19,24)(20,26,33,27)(23,34,36,30)(25,32);(1,2)(3,4)(5,7,9,18,10,19,22,32)(6,11,14,16,15,25,13,24)(8,17,20,30,21,31,33,34)(12,23,27,29,28,36,26,35);(2,4)(5,8,9,20,10,21,22,33)(6,12,14,27,15,28,13,26)(7,17,18,30,19,31,32,34)(11,23,16,29,25,36,24,35);(6,13,15,14)(11,24,25,16)(12,26,28,27)(23,35,36,29)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=fh,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,f^c=fi,h^c=hi

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, 284876);