#362915

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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

932 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)(7,16)(8,20)(9,24)(10,15)(11,18)(12,19)(13,22)(14,25)(17,31)(21,26)(23,34)(27,33)(28,35)(29,32)(30,36);(5,7)(6,11,15,27)(8,17)(9,18)(10,19)(12,14,16,24)(13,26,30,31)(20,29,34,28)(21,23)(22,32)(25,33)(35,36);(5,7)(6,12,15,16)(8,21)(9,18)(10,19)(11,24,27,14)(13,29,30,28)(17,23)(20,31,34,26)(22,35)(25,33)(32,36);(5,8,9,22,10,23,25,36)(6,13,14,20,15,30,24,34)(7,17,18,32,19,21,33,35)(11,26,16,29,27,31,12,28)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,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,d^c=di,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307574, #72307582, #72307762
maximal normal subgroup of	#72307574, #72307582, #72307762

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270111);