#426018

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1004 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)(7,11)(8,19,20,28)(9,13)(10,14)(12,21,27,22)(15,30,31,36)(16,18)(17,25)(23,29)(24,32,26,34)(33,35);(5,7,9,17,10,18,23,35)(6,11,13,25,14,16,29,33)(8,15,22,34,20,31,21,32)(12,26,28,30,27,24,19,36);(5,8,9,22,10,20,23,21)(6,12,13,28,14,27,29,19)(7,15,17,34,18,31,35,32)(11,24,25,36,16,26,33,30);(2,4)(5,6)(7,16)(8,12)(9,13)(10,14)(11,18)(15,26)(17,33)(19,21)(20,27)(22,28)(23,29)(24,31)(25,35)(30,34)(32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250492, #72252967, #72256905
maximal normal subgroup of	#72250492, #72252967, #72256905

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 333214);