#423430

order	512
nilpotency class	3
isoclinism	isoclinic class of #419466
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

1501 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)(7,11)(8,12,20,26)(9,13)(10,14)(15,22,18,30)(16,23)(17,24)(19,25,32,36)(21,27)(28,33,31,35)(29,34);(5,7)(6,11)(8,18,20,15)(9,16)(10,17)(12,22,26,30)(13,23)(14,24)(19,28,32,31)(21,29)(25,35,36,33)(27,34);(5,8)(6,12,14,26)(7,15)(9,19)(10,20)(11,22,24,30)(13,25,27,36)(16,28)(17,18)(21,32)(23,33,34,35)(29,31);(2,4)(5,9,10,21)(6,13,14,27)(7,16,17,29)(8,19,20,32)(11,23,24,34)(12,25,26,36)(15,28,18,31)(22,33,30,35)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285347, #72285428, #72285479, …
maximal normal subgroup of	#72285347, #72285428, #72285479, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330626);