#41×#131 #10574264

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

#41×#131

Subgroups

	subgroup	G/H
center	C₄×C₈	C₂⁴
commutator, socle	C₂²	C₂⁵×C₄
Frattini, cosocle	C₂×C₄	C₂⁶
prime radical	#41²	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

7085 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

(1,2)(3,6)(4,5)(7,8);(2,5)(6,8);(1,3,4,7)(2,6,5,8);(9,10,12,15,13,16,19,22)(11,14,18,21,17,20,23,24);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24);(11,17)(14,20)(18,23)(21,24)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285904, #41×#444, #72315641, …
maximal normal subgroup of	#72285904, #41×#444, #72315641, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10481460);