#411185

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

914 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265308, #72307328, #72307337
maximal normal subgroup of	#72265308, #72307328, #72307337

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318381);