#422758

order	512
nilpotency class	3
isoclinism	isoclinic class of #420461
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60644	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1981 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316858
maximal normal subgroup of	#72316858
maximal quotient of	#72316866, #72316978, #72317019

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 329954);