#354316

order	512
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1882 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250386, #72252681, #72252972, …
maximal normal subgroup of	#72250386, #72252681, #72252972, …
maximal quotient of	#72313903, C₂×#354316, #72343056, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261512);