#394025

order	512
nilpotency class	3
isoclinism	isoclinic class of #394022
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3586	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3896 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)(5,6,12,8)(7,17,22,18)(9,14,26,24)(10,16,11,15)(13,21,23,20)(19,32,34,31)(25,30,27,29)(28,35,36,33);(5,7)(6,13)(8,23)(9,19)(10,21)(11,20)(12,22)(14,28)(15,18)(16,17)(24,36)(25,35)(26,34)(27,33)(29,31)(30,32);(2,4)(5,8,12,6)(7,18,22,17)(9,24,26,14)(10,15,11,16)(13,20,23,21)(19,32,34,31)(25,29,27,30)(28,35,36,33);(5,9)(6,14)(7,19)(8,24)(10,27)(11,25)(12,26)(13,28)(15,30)(16,29)(17,31)(18,32)(20,35)(21,33)(22,34)(23,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239341, #72302583, #72302623
maximal normal subgroup of	#72239341, #72302583, #72302623
maximal quotient of	#72280739, #72316700

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301221);