#404006

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	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2539 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)(9,17)(10,15)(11,16)(12,20)(14,23)(19,27)(21,28)(22,29)(24,32)(25,31)(26,35)(30,33)(34,36);(5,7,11,22)(6,12,16,28)(8,19,24,34)(9,21,10,20)(13,30,32,26)(14,27,31,36)(15,29,17,18)(23,35,25,33);(2,4)(5,8)(6,13)(7,19)(9,25)(10,23)(11,24)(12,26)(14,15)(16,32)(17,31)(18,27)(20,35)(21,33)(22,34)(28,30)(29,36);(5,8)(6,14,16,31)(7,19)(9,25)(10,23)(11,24)(12,27,28,36)(13,15,32,17)(18,30,29,26)(20,35)(21,33)(22,34)

Polycyclic presentation:

2^9|b^2=i,d^2=g,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	#72306505, #72306508, #72306516
maximal normal subgroup of	#72306505, #72306508, #72306516

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311202);