#368401

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1314 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=g,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=di,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249949, #72249953, #72249969, …
maximal normal subgroup of	#72249949, #72249953, #72249969, …
maximal quotient of	#72265673, #72279550, #72343109

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275597);