#404007

order	512
nilpotency class	3
isoclinism	isoclinic class of #396101
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23445	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2633 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)(7,18)(8,13)(9,16)(10,17)(11,15)(12,21)(14,24)(19,33)(20,28)(22,29)(23,30)(25,32)(26,31)(27,35)(34,36);(5,7,10,22)(6,12,17,30)(8,20,25,19)(9,23,11,21)(13,27,32,36)(14,28,31,33)(15,29,16,18)(24,34,26,35);(2,4)(5,8)(6,13)(7,19)(9,26)(10,25)(11,24)(12,27)(14,15)(16,31)(17,32)(18,33)(20,22)(21,35)(23,34)(28,29)(30,36);(5,8)(6,14,17,31)(7,20)(9,26)(10,25)(11,24)(12,28,30,33)(13,15,32,16)(18,27,29,36)(19,22)(21,34)(23,35)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306379, #72306505, #72306507
maximal normal subgroup of	#72306379, #72306505, #72306507

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311203);