#147453

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	#1926
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

730 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=h,f^2=i,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	#72314250, #72314281
maximal normal subgroup of	#72314250, #72314281

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54649);