#429990

order	512
nilpotency class	3
isoclinism	isoclinic class of #419466
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#23444	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1732 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)(7,11,17,24)(8,12)(9,13)(10,14)(15,25,18,22)(16,23,31,28)(19,26)(20,27)(21,29)(30,35,32,34)(33,36);(5,7)(6,11,14,24)(8,18,20,15)(9,16,21,31)(10,17)(12,25)(13,28)(19,30)(22,27)(23,29)(26,35,36,34)(32,33);(5,8,10,20)(6,12,14,27)(7,15,17,18)(9,19,21,33)(11,22,24,25)(13,26,29,36)(16,30,31,32)(23,34,28,35);(2,4)(5,9)(6,13)(7,16,17,31)(8,19)(10,21)(11,23,24,28)(12,26)(14,29)(15,30,18,32)(20,33)(22,34,25,35)(27,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251368, #72251395, #72251401, …
maximal normal subgroup of	#72251368, #72251395, #72251401, …
maximal quotient of	#72251448

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 337186);