#417533

order	512
nilpotency class	3
isoclinism	isoclinic class of #417532
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#3026	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2626 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)(5,6,10,14)(7,11,17,23)(8,12,19,25)(9,13,20,26)(15,27,29,21)(16,22,30,34)(18,24,32,36)(28,35,31,33);(5,7)(6,11)(8,15)(9,16)(10,17)(12,21)(13,22)(14,23)(18,31)(19,29)(20,30)(24,35)(25,27)(26,34)(28,32)(33,36);(5,8)(6,12)(7,15)(9,18)(10,19)(11,21)(13,24)(14,25)(16,28)(17,29)(20,32)(22,33)(23,27)(26,36)(30,31)(34,35);(2,4)(5,9)(6,13)(7,16)(8,18)(10,20)(11,22)(12,24)(14,26)(15,28)(17,30)(19,32)(21,33)(23,34)(25,36)(27,35)(29,31)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252226, #72261220, #72268304, …
maximal normal subgroup of	#72252226, #72261220, #72268304, …
maximal quotient of	#72252217, #72261264, #72261443, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324729);