#147483

order	512
nilpotency class	4
isoclinism	isoclinic class of #13030
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

746 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)(7,12,17,28)(8,11)(9,23)(10,14)(13,21)(15,20)(16,24,32,35)(18,34,31,29)(19,26)(22,25)(27,30,36,33);(6,11,14,25)(8,19)(9,20)(12,24,28,35)(13,23,26,15)(16,30)(18,31)(21,22)(27,34,36,29)(32,33);(1,2,3,4)(5,7)(6,12,14,28)(8,16)(9,18)(10,17)(11,24,25,35)(13,27,26,36)(15,29,23,34)(19,30)(20,31)(21,33)(22,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249880, #72249884, #72249895, …
maximal normal subgroup of	#72249880, #72249884, #72249895, …
maximal quotient of	#72265700, #72279924, C₂×#147483

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54679);