Finite Group Databasebeta

← Prev (#6379146) Next (#6379148) →

C₂²⋊C₄×AGL(1,ℤ/8) #6379147

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

C₂²⋊C₄×AGL(1,ℤ/8)

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×AGL(1,ℤ/8)	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5608 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 16 points:

(1,2,4,6)(3,5,7,8);(2,5)(6,8);(9,10)(11,13)(12,14)(15,16);(10,13)(11,16)(14,15);(10,14)(13,15)

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249507, #120×AGL(1,ℤ/8), #72249558, …
maximal normal subgroup of	#72249507, #120×AGL(1,ℤ/8), #72249558, …
maximal quotient of	#72264411, #72264418, #72264502, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6286343);