Finite Group Databasebeta

← Prev (#10574272) Next (#10574274) →

C₂²×#2973 #10574273

order	512
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator	C₂²	C₂⁵×C₄
Frattini, cosocle	C₂×C₄	C₂⁶
socle	C₂⁴	C₂³×C₄
prime radical	C₂⁴×D₄	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

9978 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 20 points:

(1,2);(3,4);(5,6)(7,11)(8,10)(9,12)(13,14,15,18,16,17,19,20);(5,7)(6,9)(8,12)(10,11);(7,12)(9,11)(14,17)(18,20);(6,10)(9,11)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#60230, C₂²×#8967, C₂²×#61569, …
maximal normal subgroup of	C₂²×#60230, C₂²×#8967, C₂²×#61569, …
maximal quotient of	#72309254, #72309349, #72309352

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10481469);