#392404

order	512
nilpotency class	3
isoclinism	isoclinic class of #188764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2283
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1302 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)(7,17,21,34)(8,22,19,13)(9,14)(10,15)(11,16)(12,20,30,23)(18,35)(24,29,36,31)(25,32)(26,33)(27,28);(5,7,10,20,11,21,27,23)(6,12,28,34,16,30,15,17)(8,18,24,9,19,25,36,26)(13,14,29,35,22,33,31,32);(5,8)(6,13)(7,18)(9,20)(10,24)(11,19)(12,14)(15,31)(16,22)(17,32)(21,25)(23,26)(27,36)(28,29)(30,33)(34,35);(2,4)(5,9,10,25,11,26,27,18)(6,14,15,32,16,33,28,35)(7,19,20,36,21,8,23,24)(12,29,17,13,30,31,34,22)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,h^b=hi,d^c=di,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250125, #72250126, #72250134, …
maximal normal subgroup of	#72250125, #72250126, #72250134, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299600);