#375195

order	512
nilpotency class	3
isoclinism	isoclinic class of #213728
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1707 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)(7,15,18,11)(8,12)(9,22,23,13)(10,14)(16,28,19,24)(17,25)(20,26,33,34)(21,27)(29,31)(30,36)(32,35);(1,2,3,4)(5,7)(6,11)(8,19)(9,17)(10,18)(12,24)(13,25)(14,15)(16,21)(20,32)(22,29)(23,31)(26,35)(27,28)(30,33)(34,36);(2,4)(5,8,10,21)(6,12,14,27)(7,16)(9,20,23,33)(11,24)(13,26,22,34)(15,28)(17,30)(18,19)(25,35)(29,36)(31,32);(5,9)(6,13)(7,17)(8,20)(10,23)(11,25)(12,26)(14,22)(15,29)(16,30)(18,31)(19,32)(21,33)(24,35)(27,34)(28,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283616, #72283866, #72283938
maximal normal subgroup of	#72283616, #72283866, #72283938

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 282391);