#429698

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

909 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)(7,12)(8,19,21,32)(9,14)(10,15)(11,16)(13,22,30,25)(17,34,35,20)(18,23)(24,31)(26,33)(27,36,28,29);(5,7,10,14,11,18,26,31)(6,12,15,9,16,23,33,24)(8,17,25,29,21,35,22,36)(13,28,32,34,30,27,19,20);(5,8)(6,13)(7,17)(9,20)(10,25)(11,21)(12,27)(14,29)(15,32)(16,30)(18,35)(19,33)(22,26)(23,28)(24,34)(31,36);(2,4)(5,9,10,23,11,24,26,12)(6,14,15,18,16,31,33,7)(8,20,25,28,21,34,22,27)(13,29,32,35,30,36,19,17)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250174, #72252967, #72256865
maximal normal subgroup of	#72250174, #72252967, #72256865

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 336894);