#449582

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2404
prime radical	C₂×#3025	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2080 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)(7,15,17,27)(8,12)(9,13)(10,14)(11,18,23,21)(16,29,30,36)(19,25)(20,26)(22,28)(24,31,34,33)(32,35);(8,19)(11,23)(12,25)(15,27)(20,32)(24,34)(26,35)(29,36);(5,7)(6,11)(8,16)(9,21)(10,17)(12,24)(13,27)(14,23)(15,28)(18,22)(19,30)(20,33)(25,34)(26,36)(29,35)(31,32);(2,4)(5,8)(6,12)(7,16)(9,20)(10,19)(11,24)(13,26)(14,25)(15,29)(17,30)(18,31)(21,33)(22,32)(23,34)(27,36)(28,35)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285522, #72285534, #72285668, …
maximal normal subgroup of	#72285522, #72285534, #72285668, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356778);