C₂⁴×2₋¹⁺⁴ #10587010

order	512
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁷

Aliases

direct product

C₂⁴×2₋¹⁺⁴

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁸
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂ ▷ C₁

422866 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 8 generators on 24 points:

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

Conjugacy classes & character table

272 conjugacy classes. The character table is too large to display.

Relationships

derived subgroup of	#72266144, C₂×#72211295, #72268893
Fitting subgroup of	C₂²×#87545, #72227574, #72227575, …
maximal subgroup of	#72210333, #72210817, #72211295, …
maximal normal subgroup of	#72210333, #72210817, #72211295, …
maximal quotient of	C₂⁵×2₋¹⁺⁴, #72268062, 2₋¹⁺⁴², …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10494206);

C24×2−1+4 #10587010