#414978

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1248 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)(7,17,20,12)(8,21)(9,16)(10,27)(11,15)(13,19,29,26)(14,24)(18,30,22,31)(23,32)(25,35)(28,34,33,36);(2,4)(5,7)(6,12)(8,18)(9,26)(10,19)(11,20)(13,27)(14,28)(15,17)(16,29)(21,31)(22,25)(23,36)(24,34)(30,35)(32,33);(5,7)(6,13)(8,22)(9,26)(10,19)(11,20)(12,16)(14,31)(15,29)(17,27)(18,25)(21,33)(23,34)(24,36)(28,35)(30,32);(5,8,10,24,11,25,9,23)(6,14,16,21,15,32,27,35)(7,18,19,34,20,22,26,36)(12,28,29,31,17,33,13,30)

Polycyclic presentation:

2^9|a^2=efh,d^2=h,f^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,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	#72231480, #72231526, #72231555, …
maximal normal subgroup of	#72231480, #72231526, #72231555, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322174);