#149200

order	512
nilpotency class	4
isoclinism	isoclinic class of #13460
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2058
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

538 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=e,d^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,g^b=gi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224865, #72224931, #72226807, …
maximal normal subgroup of	#72224865, #72224931, #72226807, …
maximal quotient of	#72224857, #72237232, #72279502

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56396);