#417967

order	512
nilpotency class	3
isoclinism	isoclinic class of #417571
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3020	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1966 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)(7,15,18,11)(8,12)(9,14)(10,13)(16,19,29,22)(17,31,34,25)(20,27)(21,26)(23,30)(24,28)(32,36,33,35);(5,6)(7,16,18,29)(8,12)(9,14)(10,13)(11,19,15,22)(17,33,34,32)(20,27)(21,26)(23,30)(24,28)(25,35,31,36);(5,7)(6,11)(8,17)(9,22)(10,18)(12,25)(13,15)(14,29)(16,28)(19,24)(20,36)(21,34)(23,35)(26,31)(27,32)(30,33);(2,4)(5,8,10,21)(6,12,13,26)(7,17,18,34)(9,23,24,20)(11,25,15,31)(14,30,28,27)(16,32,29,33)(19,36,22,35)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262969, #72263055, #72263057, …
maximal normal subgroup of	#72262969, #72263055, #72263057, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325163);