C₂⁵×C₂²⋊Q₈ #72291291

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁷

Aliases

direct product

C₂⁵×C₂²⋊Q₈

Subgroups

	subgroup	G/H
center, socle	C₂⁷	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁸
prime radical	C₂⁸	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂² ▷ C₁

1048277 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 8 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂⁴×#2407, C₂⁵×#546, C₂⁵×#525, …
maximal normal subgroup of	C₂⁴×#2407, C₂⁵×#546, C₂⁵×#525, …
maximal quotient of	C₂⁴×#2848, C₂³×#36154, C₂³×#36215

GAP

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

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

C25×C22⋊Q8 #72291291