#493438

order	512
nilpotency class	4
isoclinism	isoclinic class of #480274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂×#3500	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

3382 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)(5,6,11,16,12,17,8,14)(7,18,19,34,21,29,26,36)(9,15,23,33,10,24,22,32)(13,27,30,25,31,35,28,20);(5,7)(6,13)(8,19)(9,20)(10,25)(11,26)(12,21)(14,28)(15,29)(16,30)(17,31)(18,24)(22,27)(23,35)(32,34)(33,36);(2,4)(5,8,12,11)(6,14,17,16)(7,19,21,26)(9,23,10,22)(13,28,31,30)(15,33,24,32)(18,34,29,36)(20,35,25,27);(5,9,12,10)(6,15,17,24)(7,20,21,25)(8,22,11,23)(13,29,31,18)(14,32,16,33)(19,27,26,35)(28,34,30,36)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,g^b=ghi,d^c=di,g^e=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261597, #72261611, #72261697, …
maximal normal subgroup of	#72261597, #72261611, #72261697, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400634);