#230434

order	512
nilpotency class	3
isoclinism	isoclinic class of #20158
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	#19157	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

604 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:

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

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,d^c=dh,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137630);