#360760

order	512
nilpotency class	3
isoclinism	isoclinic class of #25798
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
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

1368 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227326, #72280888, #72281024, …
maximal normal subgroup of	#72227326, #72280888, #72281024, …
maximal quotient of	#72313558

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267956);