#356233

order	512
nilpotency class	3
isoclinism	isoclinic class of #22455
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1410 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280827, #72280848, #72280870, …
maximal normal subgroup of	#72280827, #72280848, #72280870, …
maximal quotient of	#99×#106, #72313705

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263429);