#354828

order	512
nilpotency class	3
isoclinism	isoclinic class of #3019
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#379
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3612 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214887, #72224693, #72227321, …
maximal normal subgroup of	#72214887, #72224693, #72280941, …
maximal quotient of	#72281688

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262024);