#404003

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2404
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2633 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)(7,17)(8,13)(9,16)(10,14)(11,15)(12,19)(18,32)(20,26)(21,30)(22,31)(23,27)(24,29)(25,35)(28,34)(33,36);(5,7)(6,12)(8,18)(9,23)(10,19)(11,20)(13,28)(14,26)(15,27)(16,17)(21,33)(22,35)(24,34)(25,30)(29,36)(31,32);(2,4)(5,8)(6,13)(7,18)(9,24)(10,22)(11,21)(12,25)(14,31)(15,30)(16,29)(17,32)(19,35)(20,33)(23,34)(26,36)(27,28);(6,14,15,16)(8,21)(12,26,27,17)(13,29,30,31)(18,33)(22,24)(25,32,28,36)(34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250788, #72250789, #72250799, …
maximal normal subgroup of	#72250788, #72250789, #72250799, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311199);