#379702

order	512
nilpotency class	3
isoclinism	isoclinic class of #24985
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

639 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)(11,19)(14,17)(18,28)(20,26)(21,35)(22,29)(23,30)(24,31)(25,33)(27,32)(34,36);(5,7,17,32,10,20,9,19)(6,11,13,26,15,27,14,16)(8,21,22,18,24,33,23,34)(12,28,30,35,31,36,29,25);(5,8,9,22,10,24,17,23)(6,12,14,30,15,31,13,29)(7,18,19,33,20,34,32,21)(11,25,16,28,27,35,26,36);(1,3,4,2)(6,13,15,14)(11,26,27,16)(12,29,31,30)(25,36,35,28)

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,f^c=fi,g^c=gi

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, 286898);