#181457

order	512
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1634 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,e^a=eh,g^a=gh,c^b=ci,d^b=dh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240033, #72240049, #72240053
maximal normal subgroup of	#72240033, #72240049, #72240053
maximal quotient of	#72240037

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88653);