#411662

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1860 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)(7,13)(8,21)(9,17)(10,15)(11,16)(12,19)(14,23)(18,32)(20,29)(22,31)(24,30)(25,27)(26,34)(28,33)(35,36);(5,7)(6,12)(8,18)(9,25)(10,19)(11,20)(13,17)(14,26)(15,29)(16,27)(21,28)(22,35)(23,33)(24,34)(30,36)(31,32);(2,4)(5,7)(6,13)(8,18)(9,25)(10,19)(11,20)(12,15)(14,28)(16,29)(17,27)(21,32)(22,35)(23,33)(24,34)(26,30)(31,36);(5,8,10,23,11,24,9,22)(6,14,17,21,16,31,15,30)(7,18,19,33,20,34,25,35)(12,26,13,28,27,32,29,36)

Polycyclic presentation:

2^9|d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281194, #72281244, #72281398, …
maximal normal subgroup of	#72281194, #72281244, #72281398, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318858);