#497354

order	512
nilpotency class	4
isoclinism	isoclinic class of #480866
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#37654	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#546 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1272 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,g^c=gh,g^f=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312231, #72312239, #72312266, …
maximal normal subgroup of	#72312231, #72312239, #72312266, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404550);