#363975

order	512
nilpotency class	3
isoclinism	isoclinic class of #23998
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

2490 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)(7,12,19,30)(8,21)(9,14,17,34)(10,24)(11,15)(13,22)(16,27)(18,35)(20,32,25,29)(23,31,36,28)(26,33);(2,4)(5,7)(6,12,15,30)(8,17)(9,18)(10,25)(11,19)(13,31,33,28)(14,21,34,35)(16,32,24,29)(20,27)(22,23)(26,36);(5,8,11,18)(6,13,15,33)(7,17,19,9)(10,26,27,22)(12,28,30,31)(14,32,34,29)(16,35,24,21)(20,23,25,36);(2,4)(5,9,11,17)(6,14)(7,18,19,8)(10,23,27,36)(12,21)(13,32)(15,34)(16,28)(20,26,25,22)(24,31)(29,33)(30,35)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306319, #72306371, #72306379, …
maximal normal subgroup of	#72306319, #72306371, #72306379, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271171);