#371158

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	#2598
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1162 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285177, #72285600, #72285631
maximal normal subgroup of	#72285177, #72285600, #72285631

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278354);