C₂⁴×D₄×Q₈ #72339120

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

Aliases

C₂⁴×D₄×Q₈

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁸
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂² ▷ C₁

858725 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 8 generators on 20 points:

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

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

maximal subgroup of	C₂⁴×D₄×#41
maximal normal subgroup of	C₂⁴×D₄×#41

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

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