#492842

order	512
nilpotency class	2
isoclinism	isoclinic class of #20338
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₄²
commutator	C₄	C₂²×C₄×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1626 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)(5,6,9,15,10,12,23,25,11,16,22,30,8,14,21,29)(7,13,19,28,20,24,33,34,17,26,31,35,18,27,32,36);(2,4)(6,12,16,14)(9,22)(13,24,26,27)(15,29,30,25)(19,31)(21,23)(28,36,35,34)(32,33);(5,7)(6,13)(8,18)(9,19)(10,20)(11,17)(12,24)(14,27)(15,28)(16,26)(21,32)(22,31)(23,33)(25,34)(29,36)(30,35);(7,17)(13,26)(18,20)(19,31)(24,27)(28,35)(32,33)(34,36)

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=i,f^2=h,h^2=i,b^a=be,g^a=gi,f^b=fi,d^c=di

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239459, #72239465, #72264790, …
maximal normal subgroup of	#72239459, #72239465, #72264790, …
maximal quotient of	D₄×#1375

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400038);