#416373

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

994 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249838, #72249840, #72249848, …
maximal normal subgroup of	#72249838, #72249840, #72249848, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323569);