#415602

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
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	#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

1187 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)(7,12,19,29)(8,21,18,13)(9,14)(10,15)(11,16)(17,27)(20,28,25,33)(22,30)(23,34,26,31)(24,32)(35,36);(2,4)(5,7)(6,12)(8,22)(9,18)(10,19)(11,25)(13,30)(14,21)(15,29)(16,33)(17,23)(20,24)(26,35)(27,31)(28,32)(34,36);(5,8)(6,13)(7,17)(9,20)(10,18)(11,26)(12,27)(14,28)(15,21)(16,34)(19,35)(22,25)(23,24)(29,36)(30,33)(31,32);(5,9,11,17,10,22,24,35)(6,14,16,27,15,30,32,36)(7,18,25,23,19,8,20,26)(12,21,33,31,29,13,28,34)

Polycyclic presentation:

2^9|a^2=h,d^2=gh,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,f^b=fi,g^b=gi,h^b=hi,e^c=ei,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, #72257140, #72257214
maximal normal subgroup of	#72250285, #72257140, #72257214

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322798);