#362877

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1170 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)(5,6,11,16,12,17,10,15)(7,9,14,27,19,25,20,26)(8,13,24,31,23,30,28,32)(18,33,21,36,34,22,35,29);(5,7,10,20,12,19,11,14)(6,9,15,26,17,25,16,27)(8,21,24,34,23,35,28,18)(13,29,31,33,30,36,32,22);(5,8,10,28,12,23,11,24)(6,13,15,32,17,30,16,31)(7,18,20,35,19,34,14,21)(9,22,26,36,25,33,27,29);(5,9)(6,14)(7,15)(8,22)(10,26)(11,27)(12,25)(13,21)(16,19)(17,20)(18,32)(23,33)(24,29)(28,36)(30,35)(31,34)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237223, #72239887, #72239893, …
maximal normal subgroup of	#72237223, #72239887, #72239893, …
maximal quotient of	#72237241

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270073);