#147450

order	512
nilpotency class	4
isoclinism	isoclinic class of #2238
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#481
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1874
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

766 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226840, #72250249, #72250650, …
maximal normal subgroup of	#72226840, #72250249, #72250650, …
maximal quotient of	#72307594

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54646);