#147404

order	512
nilpotency class	4
isoclinism	isoclinic class of #13074
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
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

726 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)(6,12,14,15)(8,20)(9,23)(10,22)(13,27,28,24)(16,30,25,26)(17,31)(18,32)(21,33)(29,35,34,36);(5,6,9,15,10,16,23,30,11,14,8,12,22,25,20,26)(7,13,17,27,18,29,31,36,19,28,21,24,32,34,33,35);(1,2,3,4)(5,7)(6,13)(8,21)(9,17)(10,18)(11,19)(12,24)(14,28)(15,27)(16,29)(20,33)(22,32)(23,31)(25,34)(26,35)(30,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224893, #72226816, #72226819, …
maximal normal subgroup of	#72224893, #72226816, #72226819, …
maximal quotient of	#72224891

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54600);