C₂⁴×#1580 #72286612

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₈ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂⁴×#1580

Subgroups

	subgroup	G/H
center	C₂⁵×C₄	C₂×D₄
commutator	C₄	C₂⁶×C₈
Frattini, cosocle	C₄²	C₂⁷
socle	C₂⁶	C₄×D₄
prime radical	C₂⁵×D₈	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₄ ◁ C₂⁴×C₄×C₈ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

234792 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 7 generators on 24 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	#72286863, C₂⁴×#12067, #72322335
maximal normal subgroup of	#72286863, C₂⁴×#12067, #72322335
maximal quotient of	C₂⁴×#12067

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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

C24×#1580 #72286612