#147682

order	512
nilpotency class	4
isoclinism	isoclinic class of #13310
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1028 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249891, #72249912, #72249932, …
maximal normal subgroup of	#72249891, #72249912, #72249932, …
maximal quotient of	#72264805

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54878);