#230418

order	512
nilpotency class	3
isoclinism	isoclinic class of #20022
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2361
prime radical	#19125	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

635 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,6,5,7,8,3)(9,10)(11,22,28,37)(12,29)(13,18)(14,19)(15,21)(16,26,34,27)(17,31)(20,23)(24,25,36,39)(30,35)(32,40,38,33);(2,3)(4,8)(6,7)(9,11,23,26,15,28,14,27)(10,16,20,22,21,34,19,37)(12,24,31,33,18,36,30,40)(13,32,35,39,29,38,17,25);(9,12)(10,17)(11,24)(13,20)(14,30)(15,18)(16,25)(19,29)(21,35)(22,32)(23,31)(26,33)(27,40)(28,36)(34,39)(37,38);(1,3,4,2,5,6,8,7)(9,13,14,17,15,29,23,35)(10,18,19,31,21,12,20,30)(11,25,26,32,28,39,27,38)(16,33,22,36,34,40,37,24)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dhi,f^a=fi,d^b=de,e^b=eh,g^b=gh,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286191, #72286208, #72286365
maximal normal subgroup of	#72286191, #72286208, #72286365
maximal quotient of	#72287105, #72287209, #72287271, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137614);