#427388

order	512
nilpotency class	3
isoclinism	isoclinic class of #427387
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2760
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1402 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283616, #72283657, #72283682, …
maximal normal subgroup of	#72283616, #72283657, #72283682, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 334584);