#113888

order	512
nilpotency class	3
isoclinism	isoclinic class of #113886
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=di,e^b=eh,g^b=ghi,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257736, #72257749, #72263960
maximal normal subgroup of	#72257736, #72257749, #72263960
maximal quotient of	#72264382

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21084);