#379703

order	512
nilpotency class	3
isoclinism	isoclinic class of #24986
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

629 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 286899);