#392903

order	512
nilpotency class	3
isoclinism	isoclinic class of #32947
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280888, #72281182, #72281219, …
maximal normal subgroup of	#72280888, #72281182, #72281219, …
maximal quotient of	#72285976, #72313738

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300099);