#406113

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1027 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)(7,16,18,11)(8,12)(9,13)(10,14)(15,22)(17,23,30,26)(19,25,32,24)(20,28)(21,29)(27,33)(31,35,36,34);(1,2)(3,4)(5,7,9,19,10,18,22,32)(6,11,13,24,14,16,15,25)(8,17,21,31,20,30,33,36)(12,26,29,34,28,23,27,35);(2,4)(5,8)(6,12)(7,17)(9,21)(10,20)(11,23)(13,29)(14,28)(15,27)(16,26)(18,30)(19,31)(22,33)(24,35)(25,34)(32,36);(6,13,14,15)(8,20)(11,24,16,25)(12,27,28,29)(17,30)(21,33)(23,34,26,35)(31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285151, #72285571, #72285631
maximal normal subgroup of	#72285151, #72285571, #72285631

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313309);