#149158

order	512
nilpotency class	4
isoclinism	isoclinic class of #13490
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2073
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

424 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)(7,18)(8,9)(10,23)(13,30)(14,27)(15,16)(17,29)(19,33)(20,35)(21,36)(22,25)(24,26)(28,31);(5,6)(7,19,22,31)(8,14)(9,12)(10,27)(11,17)(13,18,30,25)(15,24)(16,23)(20,32,36,34)(21,28,35,33)(26,29);(5,7,9,25,10,21,24,36,11,22,26,18,23,35,8,20)(6,13,14,19,16,32,29,33,17,30,15,31,27,34,12,28)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321505, #72321571, #72321592, …
maximal normal subgroup of	#72321505, #72321571, #72321592, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56354);