#362388

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

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

904 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)(7,17)(8,22,25,13)(9,14)(10,15)(11,16)(12,20)(18,29,34,33)(19,32)(21,28)(23,36,35,27)(24,31,26,30);(1,2,3,4)(5,7)(6,12,16,28)(8,23,25,35)(9,20)(10,21)(11,19)(13,29)(14,32,15,17)(18,24,34,26)(22,33)(27,31)(30,36);(2,4)(5,8)(6,13)(7,18)(9,24)(10,26)(11,25)(12,27)(14,30)(15,31)(16,22)(17,33)(19,34)(20,23)(21,35)(28,36)(29,32);(6,14,16,15)(7,19)(12,17,28,32)(13,30,22,31)(18,34)(20,21)(23,35)(27,33,36,29)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216230, #72280083, #72280664
maximal normal subgroup of	#72216230, #72280083, #72280664
maximal quotient of	#72215774

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 269584);