#402878

order	512
nilpotency class	3
isoclinism	isoclinic class of #394465
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1157 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)(5,6,9,23,10,13,24,14)(7,15,19,26,20,11,18,16)(8,12,21,34,22,27,35,28)(17,29,32,36,33,25,31,30);(5,7)(6,11)(8,17)(9,18)(10,20)(12,25)(13,15)(14,16)(19,24)(21,31)(22,33)(23,26)(27,29)(28,30)(32,35)(34,36);(5,6)(7,16)(8,12)(9,23)(10,13)(11,18)(14,24)(15,19)(17,30)(20,26)(21,34)(22,27)(25,31)(28,35)(29,32)(33,36);(2,4)(5,8,9,21,10,22,24,35)(6,12,23,34,13,27,14,28)(7,17,18,31,20,33,19,32)(11,25,26,36,15,29,16,30)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250134, #72257150, #72257207
maximal normal subgroup of	#72250134, #72257150, #72257207

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310074);