#400969

order	512
nilpotency class	3
isoclinism	isoclinic class of #393899
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2397
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1746 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)(7,16,20,17)(8,13)(9,14)(10,25)(11,15)(12,19,27,21)(18,30,33,31)(22,28)(23,35)(24,29)(26,32,36,34);(5,7,9,21,11,20,10,19)(6,12,14,17,15,27,25,16)(8,18,22,34,24,33,23,32)(13,26,28,31,29,36,35,30);(5,6)(7,17)(8,13)(9,14)(10,25)(11,15)(12,19)(16,20)(18,31)(21,27)(22,28)(23,35)(24,29)(26,32)(30,33)(34,36);(2,4)(5,8,9,22,11,24,10,23)(6,13,14,28,15,29,25,35)(7,18,21,34,20,33,19,32)(12,26,17,31,27,36,16,30)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257146, #72257191, #72257201, …
maximal normal subgroup of	#72257146, #72257191, #72257201, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308165);