#207920

order	512
nilpotency class	3
isoclinism	isoclinic class of #161894
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

921 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)(3,6)(4,5)(7,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,19)(20,23)(21,24)(25,26)(27,30,34,39)(28,35)(29,32)(31,37)(33,40,36,38);(9,11,13,19,14,20,12,21)(10,15,16,18,17,24,22,23)(25,27)(26,30,32,39)(28,36)(29,34)(31,40,35,38)(33,37);(2,5)(6,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,24,18)(25,28,29,37)(26,31,32,35)(27,33,34,36)(30,38,39,40);(1,3,4,7)(2,6,5,8)(11,20)(15,24)(18,23)(19,21)(26,32)(27,34)(31,35)(33,36)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=fi,e^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ch,d^b=dh,e^b=ei,g^b=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 115116);