C₂²×#16512 #72321561

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂²×#16512

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#392	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

69853 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#778570, C₂²×#1393449, C₂²×#686950, …
maximal normal subgroup of	C₂²×#778570, C₂²×#1393449, C₂²×#686950, …
maximal quotient of	C₂²×#156707

GAP

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

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

C22×#16512 #72321561