#379583

order	512
nilpotency class	3
isoclinism	isoclinic class of #24986
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2572
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

964 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=efg,d^2=eh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216224, #72280622, #72280664
maximal normal subgroup of	#72216224, #72280622, #72280664
maximal quotient of	#72215774

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 286779);