#370030

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

760 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250158, #72250167, #72250174, …
maximal normal subgroup of	#72250158, #72250167, #72250174, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 277226);