#396527

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1861 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250285, #72257147, #72257218
maximal normal subgroup of	#72250285, #72257147, #72257218

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303723);